Sum-Balaban extremal dumbbell-like graph conjecture

Let $D_{a,b,a'}$ be a dumbbell graph, and assume $a\leq a'$. A dumbbell-like graph $D_{a,b,a'}^{\ell}$ is obtained from $D_{a,b,a'}$ by inserting $\ell$ edges between $v_1$ and $K_a$ when $\ell>0$, or removing $-\ell$ edges between $v_{b-1}$ and $K'_{a'}$ when $\ell<0$.

Dumbbell-like graph conjecture. Dumbbell-like graphs $D_{a,b,a'}^{\ell}$ attain the minimum value of sum-Balaban index.

The conjecture is supported by the paper's computer experiments, which examine dumbbell and dumbbell-like graphs for a range of orders. The supplied status evidence identifies it as an open problem.