Insensitivity of the minimum eigenvalue under edge addition at the optimal path-graph port

Let $n$ be odd, and let $P_n$ be a path graph with Laplacian matrix $L_n$. For 1-port selection, perturb $L_n$ to

$$\widetilde{L}_n=L_n+\epsilon e_{p^*}e_{p^*}^T,$$

where $p^*=(n+1)/2$, and denote the resulting optimal perturbed path graph by $P_n^*$. Form the disjoint union $P_n^*\cup P_n^*$, and add one edge between any pair of its nodes such that the resulting graph is connected; denote the resulting connected graph on $2n$ nodes by $T_{2n}$. Insensitivity conjecture. The minimum eigenvalues satisfy

$$\lambda_{\min}(P_n^*)=\lambda_{\min}(P_n^*\cup P_n^*)=\lambda_{\min}(T_{2n}).$$

This conjecture asserts that the minimum eigenvalue is unchanged both by taking two copies of the optimally perturbed path graph and by adding any single edge that makes the union connected. It generalizes the stated result for the path graph to all permitted pairs of nodes for the added edge; its status is unresolved in the supplied source context.