I've always felt this is the quantum equivalent of the Rayleigh-Jeans [ultraviolet] catastrophe. Observational stongly indicates the background vacuum energy state is nowhere near the inferred planckian-like density. It seems far more likely something is wrong or missing in the model. I suspect the eventual solution will explain a number of other mysteries in modern physics, like matter - antimatter abundance: which is another case where we know a tiny, positive imbalance is the outcome, but not why.

GR does not predict any density of the false vacuum, not even 10^{-120} that suggested by QM.

The hugh OOM fine tuning problem occurs when the false vacuum is identified with the cosmological constant suggested by the standard cosmological paradigm. This is an attractive hypothesis as [tex]\Lambda[/tex] and false vacuum have the same equation of state:
[tex]p=-\rho[/tex],
however strictly in GR the false vacuum ought to be entered into the right hand side of the field equation, as a component of the energy-momentum tensor of the matter field, rather than on the left hand side as a component of space-time curvature,
[tex]R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}[/tex].
i.e. [tex]\Lambda[/tex] should not be confused with T!

There is a resolution of this huge fine tuning problem, in the theory A New Self Creation Cosmology, the field equation requires a moderate false vacuum density in vacuo dependent on the curvature of space-time.

(page 712)

There are two, (the gravitational and the scalar), field equations to be satisfied in SCC. In flat space-time their solutions converge consistently, however the presence of curvature separates the solutions slightly and consistency between them requires a small false vacuum energy density. It is this that is being probed by the Casimir force.

This is testable; the Casimir force should "bottom out" in the solar gravitational field with present experimental sensitivity somewhere between the orbits of Jupiter and Saturn.

What is true of the local spherically symmetric solution is also true of the cosmological solution thereby predicting a small and precisely determined cosmological false vacuum. The cosmological solution thus requires a moderate amount of "Dark Energy" (here identified as false vacuum) [tex]\Omega_{fv}=0.11[/tex].

Black hole computers (Nov 2004)? or Loop Quantum Gravity (Jan 2004)?

There's also:
Brave New Cosmos: The Quintessential Universe (Jan 2001, part of the "Once and Future Cosmos" issue)
Exploiting Zero-Point Energy (Dec 1997)
A Cosmic Conundrum (Sep 2004, part of the "Einstein" issue)
Or perhaps from the Oct 1994 issue?

No, that's about the CMBR and its analysis (the low multipoles are observed to be quite different from what BB theory predicts, and contain some intriguing coincidences).

It is only very indirectly about this humongous inconsistency.