Twin-pair conjecture for vertex-invariant algebraic co-rank

Let $G$ be a simple graph, let $v\in V(G)$, and let $G-v$ denote the graph obtained by deleting $v$. Twin-pair conjecture. If

$$\gamma_{\mathcal P}(G-v)=\gamma_{\mathcal P}(G)\quad\text{for all }v\in V(G),$$

then $G$ has at least one pair of twin vertices. This is proposed as a stronger variant of the twin-free lower-bound conjecture and remains open.