The permanent real-rootedness conjecture

Let $A=(a_{ij})$ be a real $n\times n$ matrix with non-negative entries, and let $J$ denote the all-ones matrix. Assume that the entries of $A$ are weakly increasing down columns. The permanent real-rootedness conjecture. The polynomial

$$\operatorname{per}(A+xJ)$$

has only real roots. This conjecture is attributed in the source to Haglund; its resolution is not discussed there.