Minimal-diagram connected-sum crossing-number conjecture for links in thickened surfaces

Let $L_1\subset\Sigma_1\times I$ and $L_2\subset\Sigma_2\times I$ be links in thickened surfaces, and let $D_1,D_2$ be minimal-crossing representatives of $L_1,L_2$, respectively. Let $\Sigma_1\#\Sigma_2$ be the connected sum of the surfaces, and let $L$ be any link in its thickening arising as a connected sum of $D_1$ and $D_2$. Minimal-diagram connected-sum conjecture. Then

$$c(L)=c(L_1)+c(L_2).$$

The claim is a proposed generalization of classical crossing-number additivity. The paper shows that additivity can fail arbitrarily badly when nonminimal diagrams are used in forming the connected sum; whether it holds for connected sums of minimal crossing representatives is left as a plausible generalization.