Diagonal reduction conjecture for strong commutativity preservers of incidence algebras

Let $X$ be a finite connected poset, let $I(X,K)$ be its incidence algebra over $K$, and let $D(X,K)$ denote the diagonal subalgebra. A map $\theta:I(X,K)\to I(X,K)$ is a strong commutativity preserver if it preserves commutativity in both directions, and a map of shift type is a map $S_\alpha$ associated with a suitable map $\alpha:I(X,K)\to D(X,K)$. Let $\varphi:I(X,K)\to I(X,K)$ be a bijective strong commutativity preserver. Diagonal reduction conjecture. There exist a bijective strong commutativity preserver of shift type $S_\alpha$ and an inner automorphism $\psi$ of $I(X,K)$ such that

$$\psi\circ S_\alpha\circ\varphi$$

preserves $D(X,K)$. This would extend the preceding two-element example to arbitrary finite connected posets, reducing the study of strong commutativity preservers to those preserving the diagonal subalgebra; whether this reduction always holds is left open in the source.