Lackenby's Heegaard gradient conjecture

Let $M$ be a compact hyperbolic $3$-manifold, let $\Gamma=\pi_1(M)$, and let $\mathcal{L}$ be a family of finite-index subgroups with corresponding covers $M_i$. Define the infimal Heegaard gradient by

$$\chi^h_{\mathcal{L}}(M)=\inf_i\left\{\frac{2g(M_i)-2}{[\Gamma:N_i]}\right\}.$$

Lackenby's Heegaard gradient conjecture. If $\chi^h_{\mathcal{L}}(M)=0$ for a family of covers, then $M$ virtually fibres over a circle. The source motivates this by noting that virtual fibering gives vanishing Heegaard gradient, and conjectures that this is the only reason for vanishing.