Ayyer–Klee–Schilling conjecture on linear eigenvalues of promotion matrices

Let $P$ be a poset of size $n$ that is not a down forest, and let $M^{P}$ be its promotion transition matrix. For each element $x$ of $P$, call an element $y$ a successor of $x$ when $x\prec y$. Suppose that every eigenvalue of $M^{P}$ is linear in the parameters $x_{1},\ldots,x_{n}$. Ayyer–Klee–Schilling conjecture. Then: (1) the coefficients of the parameters in the eigenvalues are all in $\{-1,1\}$; (2) every element of $P$ has at most two successors; and (3) the only parameters whose coefficients in the eigenvalues are $-1$ are those corresponding either to elements with two successors or to elements having a successor with two successors. This conjecture seeks to classify non-down-forest posets whose promotion matrices have linear eigenvalues; the stated necessary conditions are motivated by examples such as the rank-two ladder, but the source provides no resolution of the conjecture.