Ferri–Gagliardi–Grasselli's additivity conjecture for regular genus

Let $M$ and $N$ be closed simply-connected PL $4$-manifolds, and let $\mathcal{G}(M)$ denote the graph-defined PL invariant regular genus of $M$. Ferri–Gagliardi–Grasselli's conjecture. The regular genus is additive with respect to connected sum:

$$\mathcal{G}(M\#N)=\mathcal{G}(M)+\mathcal{G}(N).$$

The conjecture is equivalent in this setting to the question of whether every closed simply-connected PL $4$-manifold admits a weak simple crystallization. A positive answer would imply universality of special handlebody decompositions and, through the relation between regular genus and the second Betti number, consequences for exotic copies of $\mathbb{S}^4$ and $\mathbb{CP}^2$.