Spider leaf-deletion conjecture for domination-polynomial modes

Let $S$ be a spider, meaning a tree with one vertex of degree greater than $2$, whose legs all have length at most $3$. Let $S'$ be obtained from $S$ by deleting a leaf, and let $D(S,x)$ and $D(S',x)$ be their domination polynomials. A mode of a unimodal polynomial is an index at which its coefficient sequence attains a maximum. Spider leaf-deletion conjecture. Both $D(S,x)$ and $D(S',x)$ are unimodal and have modes $\mu_S$ and $\mu_{S'}$, respectively, such that

$$|\mu_S-\mu_{S'}|\leq 1.$$

The paper notes that this conjecture, together with its path-recursion lemma, would imply unimodality for all spider graphs. The supplied text does not establish the conjecture.