Additivity conjecture for the regular genus under connected sum

Let $M^n_1$ and $M^n_2$ be two closed orientable $n$-manifolds. The regular genus additivity conjecture.

$$\mathcal G(M^n_1 \# M^n_2)=\mathcal G(M^n_1)+\mathcal G(M^n_2).$$

Regular genus is known to be subadditive under connected sum, and the equality is immediate in dimension three from the additivity of Heegaard genus. The conjecture remains open in general, with particular significance for closed orientable $4$-manifolds.