Lin–Zhou's distance spectral extremal conjecture

Let $m\geqslant 3$, and define

$$n:=\Bigl\lceil \tfrac{1+\sqrt{8m+1}}{2}\Bigr\rceil, \qquad s:=m-\binom{n-1}{2}\in\{1,2,\ldots,n-1\}.$$

For positive integers $N\geqslant c\geqslant 1$, let

$$P_{N,c} \,:=\,\bigl(c+c\lfloor N/c\rfloor - N\bigr)\,P_{\lfloor N/c\rfloor}\;\cup\;\bigl(N-c\lfloor N/c\rfloor\bigr)\,P_{\lceil N/c\rceil},$$

the balanced disjoint union of $c$ paths on $N$ vertices. Let $\mathcal G(m)$ be the class of connected graphs with $m$ edges, and let $\rho(G)$ denote the distance spectral radius of a connected graph $G$.

Lin–Zhou's conjecture. If $1\leqslant s\leqslant \frac{n-7}{2}$ and $G\in\mathcal G(m)$, then

$$\rho(G)\geqslant\rho\bigl(\overline{P_{n,s+1}}\bigr),$$

with equality if and only if $G\cong\overline{P_{n,s+1}}$.

Lin and Zhou proved the same extremal characterization when $\max\{\frac{n-6}{2},1\}\leqslant s\leqslant n-1$; the conjecture concerns the remaining small-$s$ range and was subsequently posed in their concluding remarks.