H-cofibrancy conjecture for regular directed realizations of precubical sets

Let $K$ be a precubical set, and let $|K|^t_{reg}$ denote its regular multipointed $d$-space realization. A multipointed $d$-space is called h-cofibrant when it is cofibrant in the relevant h-model structure. H-cofibrancy conjecture. For every precubical set $K$, the multipointed $d$-space $|K|^t_{reg}$ is h-cofibrant.

This conjecture is motivated by examples suggesting that h-cofibrant objects are precisely those without algebraic relations. Its status is not resolved in the supplied source context.