The v-number conjecture for cycles and binary trees

Let $C_n$ be the cycle graph on $n$ vertices and let $B_n$ be the binary tree of level $n$. For each graph $G$, let $J_G$ denote its binomial edge ideal.

Conjecture for cycles and binary trees. The following equalities hold:

$$\mathrm{v}(J_{C_n})=\left\lceil\frac{2n}{3}\right\rceil\quad\text{for all }n\geq 6,$$

and

$$\mathrm{v}(J_{B_n})=2^{n-1}+\mathrm{v}(J_{B_{n-3}})\quad\text{for all }n\geq 3.$$

The cycle equality would sharpen the established upper bound, while the binary-tree recurrence is suggested by preceding results and Macaulay2 computations; both assertions are presented as conjectural.