One proposal for solving the liberal dilemma is a local solution: acknowledging that it is impossible to ground a liberal government without previously adopting a value system, one instead takes the intersection of all local value systems and uses this "core" for the foundations. The result is a kind of bounded liberalism: any value system is permitted so long as it conforms to the minimal set of values in the "core". Two remarks: 1) the local solution is the approach which in practice guides most purportedly liberal societies in the modern world; 2) even a supposedly general liberal theory must prohibit some value systems (in particular, those which involve a systematic infringment on the practice of alternate value systems). Nevertheless, there are some nagging problems to the local solution which render it unsatisfactory as a foundation for liberalism.
Theoretically Determined "Core" ~ Denial of Pluralism?
If one attempts to find the intersection of local value systems through purely theoretical means (armchair political philosophy), one runs into the danger of simply imposing one's own value system. Arguably, this is the fundamental flaw in Rawls' project. John Rawls attempted to justify a liberal society using a notion of overlapping consensus to play the role of "core" mentioned above. Unfortunately, all Rawls' attempts to analyze the content of this core amount to him simply stating his own value system (as in the argument from the original position), or are simply circular (circularity being such a problem for Rawls he attempted to defend it with the notion of reflective equilibrium). Once one relativizes the liberal project to local mores, one cannot stay true to the liberal spirit by purporting to determine these local mores without empirical confirmation.
Empirically Determined "Core" ~ Mob Rule?
The only way democracy succeeds in warding off accusations of "mob rule" is by appealing to some foundational guarantee. Yes, the majority does impose it's will on the minority, but the Bill of Rights (or Magna Carta, or whatever) guarantees that this imposition will not extend to the prevention of the minority from pursuing its own value systems. Thus, democracy is only a liberal form of government if the scope of democratic choice is restricted in such a way as to protect the plurality of value systems. Yet, if the intersection of value systems which grounds the liberal society is to be determined empirically (i.e. by vote), this initial democratic act will lie outside the protections of an established society. In other words, if the "core" is not large enough to provide the appropriate basis for a government, one will be tempted to include value judgments which are held by the majority (though not with unanimity). Such a move would indeed amount to mob rule and ground a supposedly liberal society on an act of persecution. Of course, if there is enough consensus within a community, this will not be a problem. This explains why the local solution is more suited to the City-States of ancient times than to sprawling, modern "liberal" societies such as the U.S. or the E.U.
Can an Empirically Determined "Core" be Stable?
Suppose one's "core" values exhibit enough overlap to establish a liberal society. These have been determined by an initial empirical test, and have now been used to set a government in motion. Since, however, we have acknowledged that these were not absolute values, but merely the values of a particular society at a particular place and time, how do we know these values won't change over time? Does this society run the risk of oppressing itself in the future? What about immigrants and future generations ~ what is their status if they do not share the values in the justificatory "core"? One strategy for combatting this worry is that of constant ratification. The foundational documents (e.g. constitution) must be re-ratified at regular intervals. Such a procedure would ensure that at any given time the "core" value system appropriately represents the intersection of individual value systems in the society (and some Libertarians have argued for such a procedure for just this reason); nevertheless, it should be obvious that this approach is too impractical (and dangerous!) to actually implement.
However, supposing even that the local value systems remain stable, there is still a profound difficulty for this approach: the momentum of government ensures that the government will eventually encroach on aspects of life outside those dictated by the "core," and thus infringe on the value systems of the citizens. Of course, this reality of government behavior is a problem for any liberal society. One wonders, however, whether a solution to the liberal dilemma which purports to be absolute rather than contingent (as the local solution is) might have a better chance of preventing this growth of government. A society which acknowledges its government is founded merely upon contingent values is in danger of believing its changing value systems to be the cause of government change, when in fact it has misconstrued the order of causality. This is precisely what has occurred in the US: many citizens perceive the government as evolving in conformity to their demands, when actually the government has evolved in a predictable and predetermined manner, and it is the expectations of the citizens which have evolved in conformity to it.
Even as a practical measure, then, the local solution is far from satisfactory; nevertheless, it is thus far the best approach on offer.