Sink-independence conjecture for recurrent-configuration class sums

Let $G=(V,E)$) be a strongly connected digraph and let $s$ be a vertex of $G$. Write $G_{\backslash s^{+}}$ for the digraph obtained by removing all outgoing arcs of $s$, so that $s$ is a global sink. Fix an order $v_1\prec v_2\prec\cdots\prec v_{n-1}$ on the vertices distinct from $s$, where $n=|V|$, and define $r_i\in\mathbb{Z}^{n-1}$ by $r_{i,j}=\deg_G(v_i,v_j)$ for $i\ne j$ and $r_{i,i}=\deg_G^+(v_i)$. Set $\beta_i=\deg_G(s,v_i)$, let $\mathcal{C}$ be the recurrent configurations of $G_{\backslash s^{+}}$, and define $c_1\sim c_2$ when $c_1-c_2$ lies in the subgroup $\langle r_1,r_2,\ldots,r_{n-1},\beta\rangle$ of $(\mathbb{Z}^{n-1},+)$. For $B\in\mathcal{C}/\!\sim$, let

\textbf{\texttt{\sum}}\,_{G,s}(B)=\max\left\{\deg_G^+(s)+\sum_{v\ne s}c(v):c\in B\right\}.

Sink-independence conjecture. The sequence \left(\textbf{\texttt{\sum}}\,_{G,s}(B)\right)_{B\in\mathcal{C}/\sim} is independent of the choice of $s$, up to a permutation of its entries. If $G$ is Eulerian, then $\beta\in\langle r_1,\ldots,r_{n-1}\rangle$, so the equivalence relation simplifies accordingly. If true, this would generalize the specialization $T_G(1,y)$ of the Tutte polynomial from undirected graphs to strongly connected digraphs.