The odd-diameter almost seesaw conjecture for the maximum second Steklov eigenvalue

Let $D=2r+1$ be an odd integer and let $n\geq D+1$ be an integer. Write $\widetilde{\sigma}_{2,\text{max}}(D,n)$ for the maximum second Steklov eigenvalue among trees with diameter $D$ and $n$ vertices, and let $C^-(r,b,c)$ denote the lower-bound expression associated with the almost seesaw graph $AS(r,b,c)$. An almost seesaw graph $AS(r,b,c)$ is the tree obtained from the construction with parameters $r,b,c$.

Odd-diameter almost seesaw conjecture.

$$\widetilde{\sigma}_{2,\text{max}}(D,n)=C^-(r,n-2r-2,1).$$

The tree attaining the bound is the almost seesaw graph

$$AS(r,n-2r-2,1).$$

This claim identifies the extremal tree and the exact maximum for odd diameter. The supplied text gives no evidence that the statement has been proved or refuted, so its status remains open.