Beaton–Brown conjecture on the modes of domination polynomials of trees

Let $T$ be a tree and let $v\in V(T)$ be a leaf. Let $D(T)$ and $D(T\backslash v)$ denote the domination polynomials of $T$ and the tree obtained by deleting $v$, respectively. Assume that both polynomials are unimodal, and let their modes be the largest indices at which their coefficient sequences attain the required increase-then-decrease transition. Beaton–Brown's conjecture. The modes of $D(T)$ and $D(T\backslash v)$ are at distance at most $1$.

This conjecture proposes a local stability property for domination-polynomial modes under deletion of a leaf. The source presents it as an open direction for graphs such as trees, where the paper's techniques do not resolve unimodality in general.