Commutativity preservers of incidence algebras

Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $\phi :I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $\phi (D(X,K))=D(X,K)$. We prove that such a map $\phi$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(\theta ,\sigma ,c,\kappa )$ of simpler maps $\theta$, $\sigma$, $c$ and a sequence $\kappa$ of elements of $K$.