Asymptotic dumbbell conjecture for the minimum sum-Balaban index

Let $D_{a,b,a'}$ be a dumbbell graph on $n$ vertices, with clique sizes $a$ and $a'$ and connecting path size $b$, so that $a+b+a'=n$. Let

$$c=\sqrt[4]{\sqrt 2\log\big(1+\sqrt 2\big)}.$$

Asymptotic dumbbell conjecture. Among all dumbbell graphs $D_{a,b,a'}$ on $n$ vertices, the minimum is achieved for one with

$$a=c\sqrt n+o(\sqrt n),\qquad a'=c\sqrt n+o(\sqrt n),\qquad b=n-o(n).$$

This conjecture gives the predicted asymptotic shape of an optimal dumbbell graph: the two cliques have equal asymptotic size, while the path contains almost all vertices. Its status is not resolved in the supplied text.