Language compression for probabilistic time-bounded Kolmogorov complexity

Establish whether probabilistic time-bounded Kolmogorov complexity pK^t admits the language-compression property that, for every set of witnesses A_x and every witness y in A_x, one has pK^{poly(n)}(y\mid x) \leq \log |A_x|+O(\log n).

Background

The paper needs a compression theorem for probabilistic time-bounded Kolmogorov complexity in order to sample accepting witnesses for NP instances efficiently. The desired property would bound the conditional complexity of every witness by approximately the logarithm of the number of available witnesses, up to logarithmic overhead.

The authors explain that the analogous property is known for the nondeterministic variant pnK^t and can be transferred to pK^{t,SAT}, but this does not establish the property for the unrelativized pK^t measure. Resolving this question would clarify whether direct language compression is available for probabilistic time-bounded Kolmogorov complexity itself.