Kar–Nikolov's deficiency stabilisation conjecture

Let $G$) be a residually finite finitely presented group. For a finitely presented group $G$, its deficiency $\delta(G)$ is the maximum of $|X|-|R|$ over all presentations $G=\langle X\mid R\rangle$. Kar–Nikolov's deficiency stabilisation conjecture. If

$$\delta(H)-1=[G:H](\delta(G)-1)$$

for every finite-index subgroup $H$ of $G$, then $G$ has a finite $2$-dimensional classifying space $K(G,1)$. The conjecture is related to Wall's D2 Problem and the Relation Gap problem; the pro-$p$ version and higher-dimensional abstract analogues are verified, while the stated discrete conjecture is not resolved here.