Synchronizability of products of at least three cornered DFAs

Let $k\geq 3$, let $\Sigma$ be an alphabet, and let $f:\Sigma^*\rightarrow\{-1,1\}$ satisfy $f(\varepsilon)=1$. For DFAs $M_1,\ldots,M_k$ over $\Sigma$, suppose that each $M_i$ has an $f$-corner $z_i$. Their direct product is

$$M=M_1\times\cdots\times M_k.$$

A word is $(z_1,\ldots,z_k)$-synchronizing if it maps every product state to $(z_1,\ldots,z_k)$.

Product-corner conjecture. If $M_1,\ldots,M_k$ are distinct and each contains an $f$-corner $z_i$, then $M$ is $(z_1,\ldots,z_k)$-synchronizable. The conjecture extends the paper's product results from two factors to at least three factors. The source supplies no resolution, so it remains open.