The height additivity conjecture for knots

Let $K_1$ and $K_2$ be knots, and let $ht(K)$ denote the maximum number of thick spheres among all thin positions of $K$.

Height additivity conjecture. It holds that

$$ht(K_1\# K_2)=ht(K_1)+ht(K_2).$$

This conjecture concerns the behavior of the height invariant under connected sum; its status is not resolved in the supplied text.