Balanced dumbbell conjecture for the minimum sum-Balaban index

A dumbbell graph $D_{a,b,a'}$ consists of two cliques of sizes $a$ and $a'$ joined by a path with $b$ vertices, so that $a+b+a'=n$. The sum-Balaban index is denoted by ${\rm SJ}(D)$.

Balanced dumbbell conjecture. Among all dumbbell graphs $D_{a,b,a'}$ on at least $14$ vertices, the minimum value of sum-Balaban index is achieved for one with $a'=a$ or $a'=a+1$.

The restriction to at least $14$ vertices is motivated by the fact that $D_{2,7,4}$ has the lowest sum-Balaban index among dumbbell graphs on $13$ vertices. The conjecture formalizes the expectation that an optimal dumbbell graph is balanced.