Conjecture on linear factorization for posets contained in $A_k\oplus A_2$

Let $A_i$ denote an antichain of size $i$, and let $P$ be a poset whose Hasse diagram is contained in the Hasse diagram of $A_k\oplus A_2$. Let $M^{P}$ be the promotion matrix of $P$. Linear-factorization conjecture. The characteristic polynomial of $M^{P}$ factors into linear terms. This proposes a further family of posets with linear characteristic polynomials beyond the posets treated earlier in the paper; the source gives no proof or resolution.