The shifted-complex polyhedral product wedge decomposition conjecture

Let $K$ be a shifted simplicial complex, let $(CX_i,X_i,x_i)$ be based pairs indexed by the vertices of $K$, and write $Z(K;(\underline{CX},\underline{X}))$ for the associated polyhedral product. For a subset $I$ of the vertices, let $K_I$ denote the full subcomplex on $I$, let $\widehat{X}^I$ denote the smash product of the spaces $X_i$ for $i\in I$, and let $|K_I|*\widehat{X}^I$ denote their join. Shifted-complex wedge decomposition conjecture. The spaces

$$Z(K;(\underline{CX},\underline{X}))\quad\text{and}\quad\bigvee_{I\notin K}|K_I| *\widehat{X}^I$$

are of the same homotopy type. This would extend the preceding suspension decomposition to an unsuspended homotopy equivalence for shifted complexes; related unsuspended wedge-of-spheres results are known in special cases, but the stated general assertion is presented as a conjecture here.