The spectral-radius conjecture for extremal graphs with edge-disjoint spanning trees

Let $k$ and $c$ be integers with $0\le c\le k-1$. For $0\le c\le k-2$, let $\mathcal{H}_1(n,k,c)$ be the class of all $(k+c)$-edge-connected graphs $H$ of order $n$ for which there is a partition $V(H)=U\cup T$ such that $|U|=n-2c-2$, $|T|=2c+2$, $H[U]\cong K_{n-2c-2}$, $2c^2+2c+1\le e_H(T)\le 2c^2+3c+1$, and $e_H(T,U)=k(2c+2)-1-e_H(T)$. Let $H_1^*(n,k,c)$ maximize the spectral radius in $\mathcal{H}_1(n,k,c)$. For $c=k-1$, let $\mathcal{H}_2(n,k,c)$ be the class of all $(k+c)$-edge-connected graphs $H$ of order $n$ for which there is a partition $V(H)=U\cup T$ such that $|U|=n-2c-3$, $|T|=2c+3$, $H[U]\cong K_{n-2c-3}$, $e_H(T)=2c^2+3c+1$, and $e_H(T,U)=2c+1$. Let $H_2^*(n,k,c)$ maximize the spectral radius in $\mathcal{H}_2(n,k,c)$. Spectral-radius extremal conjecture. Let $n$ be sufficiently large. If $0\le c\le k-2$ and $G$ is a $(k+c)$-edge-connected graph of order $n$ with $\rho(G)\ge\rho(H_1^*(n,k,c))$, then $\tau(G)\ge k$ unless $G$ is a graph in $\mathcal{H}_1(n,k,c)$ with maximum spectral radius. If $c=k-1$ and $\rho(G)\ge\rho(H_2^*(n,k,c))$, then $\tau(G)\ge k$ unless $G$ is a graph in $\mathcal{H}_2(n,k,c)$ with maximum spectral radius. The conjecture proposes sharp spectral-radius thresholds guaranteeing $k$ edge-disjoint spanning trees, with the displayed candidate classes as the exceptional extremal graphs. The discussion is explicitly heuristic and does not prove the assertion; the cases $c=0$ and $(k,c)=(2,1)$ agree with the paper's previously established extremal results, while the general cases remain open.