Lackenby's arbitrary rank-gradient values conjecture

Let $RG(G)$ denote the absolute rank gradient of a finitely generated group $G$, defined by

$$RG(G)=\inf_{[G:H]<\infty}\frac{d(H)-1}{[G:H]},$$

where $d(H)$ is the minimum number of generators of $H$ and the infimum ranges over all finite-index subgroups $H$ of $G$.

Lackenby's conjecture. For every real number $\alpha>0$ there exists a finitely generated group $\Gamma$ such that

$$RG(\Gamma)=\alpha.$$

This conjecture asks which positive real numbers can occur as absolute rank gradients of finitely generated groups. The paper's abstract states that arbitrary positive values are constructed for the $p$-gradient, but the supplied context does not establish whether this absolute rank-gradient conjecture is thereby resolved.