Additivity conjecture for the bad-domain invariant of non-alternating knots

Let $K_1$ and $K_2$ be non-alternating knots. For a knot $K$, let $\beta(K)$ denote the minimum number of bad domains over diagrams representing $K$, and let $K_1\# K_2$ denote their connected sum. Connected-sum conjecture.

$$\beta(K_1\# K_2)=\beta(K_1)+\beta(K_2)-2.$$

The preceding construction gives the corresponding upper bound by taking connected sums of diagrams at bad edges; the conjecture asserts that this bound is always sharp for non-alternating knots. It is motivated by the additivity of knot Floer homology thickness, while the behavior of $\beta$ under connected sum is otherwise unclear.