Vertex-contraction ideal conjecture for the Tutte polynomial

Let $G$ be a connected undirected graph, let $s$ be a vertex of $G$, and let $N$ be the set of neighbors of $s$, excluding $s$. For each nonempty subset $W\subseteq N$, let $\overline{G_{/W\cup\{s\}}}$ denote the graph obtained by contracting $W\cup\{s\}$ in $G$ and removing all loops. Vertex-contraction ideal conjecture. The Tutte polynomial $T_G(x,y)$ belongs to the ideal generated by

$$\left\{T_{\overline{G_{/W\cup\{s\}}}}(x,y):\emptyset\subsetneq W\subseteq N\right\}$$

in $\mathbb{Q}[x,y]$. This conjecture would support the existence of a two-variable generalization of the Tutte polynomial for Eulerian, and possibly general, digraphs via recursive formulas involving vertex contraction.