Rank-gradient versus Heegaard-gradient conjecture

Let $M$ be a finite-volume hyperbolic $3$-manifold and let $\{M_i\longrightarrow M\}$ be a family of finite-sheeted covers. Write $\operatorname{rgr}(M,\{M_i\})$ for the rank gradient and $\operatorname{Hgr}(M,\{M_i\})$ for the Heegaard gradient of the family. Rank-gradient versus Heegaard-gradient conjecture. One has

$$\operatorname{rgr}(M,\{M_i\})>0$$

if and only if

$$\operatorname{Hgr}(M,\{M_i\})>0.$$

The source describes this as an important conjecture that would follow from the rank versus Heegaard genus conjecture. It is presented as open; the paper also notes that positive rank gradient implies positive Heegaard gradient, so the converse is the substantive direction.