Extension of Theorem 3 to higher-genus regular level sets

Determine whether the classification in Theorem 3—asserting that a 3-dimensional closed, connected, orientable manifold admits a Morse function whose regular level sets are disjoint unions of spheres S^2 and tori S^1 × S^1 if and only if the manifold is diffeomorphic to a connected sum of copies of S^1 × S^2 and a finite number of lens spaces of Heegaard genus 1—extends to the case where the regular level sets are disjoint unions of closed surfaces that may have genus greater than 1; specifically, ascertain whether an analogous classification holds for manifolds admitting Morse functions whose regular level sets include components of genus > 1.

Background

The paper’s Theorem 3 (citing Saeki’s Theorem 6.5) establishes a classification of 3-dimensional closed, connected, orientable manifolds that admit Morse functions whose regular level sets are disjoint unions of S^2 and S^1 × S^1, identifying them precisely as connected sums of copies of S^1 × S^2 and lens spaces of Heegaard genus 1.

Earlier in Section 3, the author poses whether these studies can be extended to cases where the connected surface components of regular level sets may have genus not equal to 0 or 1. Theorem 5 (Saeki’s Lemma 6.6) ensures the existence of simple Morse functions under a bound on genus but does not provide the sought classification. The author then explicitly states that it is unknown whether Theorem 3 extends to such higher-genus cases.