Perrot–Pham sink-independence conjecture for the Biggs–Merino polynomial

Let $G$ be a strongly connected digraph, and let $s$ be a sink vertex. For the set of $s$-recurrent configurations modulo the relation $\stackrel{s}{\sim}$, define the Biggs–Merino polynomial by

$$\mathcal B(G,s;y):=\sum_{[\widehat{\mathbf{c}}]\in\operatorname{Rec}(G,\stackrel{s}{\sim})}y^{\operatorname{dlvl}([\widehat{\mathbf{c}}])}.$$

Perrot–Pham's conjecture. The polynomial $\mathcal B(G,s;y)$ is independent of the choice of the sink vertex $s$.

Perrot and Pham proved this independence for connected Eulerian digraphs and conjectured it for all strongly connected digraphs. The paper states that its main theorem answers this conjecture, so the conjecture is solved.