Superbridge index additivity under connected sum with a trefoil

Let $K$ be a nontrivial knot, let $T$ be a trefoil knot, and let $K\sharp T$ denote their connected sum. Write $s[K]$ for the superbridge index of $K$.

Trefoil connected-sum conjecture. Every nontrivial knot $K$ satisfies

$$s[K\sharp T]=s[K]+1.$$

A known upper bound gives $s[K\sharp T]\le s[K]+1$ when $K$ is a torus knot. The conjecture asserts equality for every nontrivial knot and is stated to imply the conjecture on infinitely many odd-superbridge knots; the paper notes that it holds when $K$ is a trefoil or figure-eight knot.