The v-number conjecture for crown graphs

Let $G=C_{n,n}$ with $n \ge 4$. The graph $G$ is the crown graph obtained from the complete bipartite graph on two parts of size $n$ by deleting a perfect matching, and $J_G$ denotes its binomial edge ideal. For $T \in \mathcal{C}(G)$, let $P_T(G)$ be the corresponding prime ideal and let $\operatorname{v}_{P_T(G)}(J_G)$ denote the local v-number of $J_G$ at $P_T(G)$. The v-number conjecture for crown graphs. For every $T \in \mathcal{C}(G)$,

$$\operatorname{v}_{P_T(G)}(J_G)=4.$$

This would establish the expected upper value in the bound $3\leq \operatorname{v}(J_G)\leq 4$ for crown graphs with $n\geq 4$; the case $T=\emptyset$ is already known, while the assertion for every cut-set prime remains open.