The infinite-value conjecture for Perelman's invariant on a homeomorphism type

Let a homeomorphism type of smooth four-manifolds be suitable if it admits infinitely many distinct smooth structures admitting symplectic forms. The Perelman invariant takes on infinitely many distinct values on the smooth structures in a suitable homeomorphism type.

Infinite-value conjecture. There is a suitable homeomorphism type on which Perelman's invariant takes on infinitely many distinct values.

The preceding construction gives arbitrarily large finite numbers of distinct values among homeomorphic four-manifolds, but not infinitely many. The question is left open because the available estimate does not appear strong enough to establish infinitely many values for the known examples.