The handle-number bound for components of maximal surface decompositions

Let $M$ be a closed, irreducible $3$-manifold and let $\mathcal F$ be a maximal embedded collection of orientable, incompressible surfaces. If $M-N(\mathcal F)$ has $n$ components, write $M_1,\ldots,M_n$ for the component manifolds, and let $h(M_i)$ denote the handle number of $M_i$. The handle-number bound. One should have

$$\sum_{i=1}^n h(M_i) \leq g(M)+n-2.$$

This would extend the genus-two result to arbitrary closed, irreducible $3$-manifolds and provide a general upper bound on the handle numbers of component manifolds arising from a maximal surface decomposition. The source presents it as a conjectural consequence of removing the handle-number hypothesis from a preceding proposition; its status is not established here.