Heegaard genus versus discrete Morse critical edges

Let $M$ be a closed $3$-manifold, let $g$ be its Heegaard genus, and consider discrete Morse functions on triangulations of $M$; a critical edge is a critical cell of dimension one.

Heegaard–discrete Morse conjecture. Some closed $3$-manifold with Heegaard genus $g$ admits discrete Morse functions with less than $g$ critical edges.

The conjecture is motivated by examples where Heegaard genus exceeds the rank of the fundamental group. The source proposes triangulating such examples to test whether discrete Morse theory can improve on the corresponding smooth Morse bound, but gives no resolution.