Properness conjecture for commuting maps on incidence algebras

Let $(X,\leqslant)$ be a locally finite pre-ordered set, let $\mathcal{R}$ be a $2$-torsion-free commutative ring with unity, and let $I(X,\mathcal{R})$ denote the incidence algebra of $X$ over $\mathcal{R}$. For each connected component of $X$, let $\mathfrak{D}$ be the associated set, with equivalence relation $\cong$. A map $\theta:I(X,\mathcal{R})\to I(X,\mathcal{R})$ is proper if it has the form $\theta(f)=\lambda f+\mu(f)$, where $\lambda\in\mathcal{R}$ and $\mu$ is a central-valued additive map. Properness conjecture. If the set $\mathfrak{D}$ associated with each connected component forms one equivalent class under the relation $\cong$, then every commuting map of $I(X,\mathcal{R})$ is proper. The preceding theorem establishes the claim when $X$ is connected; the conjecture extends this to locally finite pre-ordered sets with finitely or arbitrarily many connected components, and the general case is left open.