Weakly homogeneous forcing changing the cofinality of the first inaccessible

A forcing notion is weakly homogeneous if its forcing relation is invariant under automorphisms of the forcing, and a forcing adds bounded subsets when the generic extension contains new subsets of ordinals below the relevant cardinal. The cofinality of a cardinal is the least order type of an unbounded subset of it.

Consistency conjecture. It is consistent, relative to the existence of large cardinals, that there is a weakly homogeneous forcing which changes the cofinality of the first inaccessible to $\omega$ without adding bounded subsets.

This conjecture asks whether the obstruction established for changing the cofinality of the first inaccessible to $\omega_1$ can be avoided when the target cofinality is $\omega$. The required large-cardinal strength is not specified here, and the consistency claim remains open in the supplied text.