Ivanov's presentation conjecture for power subgroups of mapping class groups

Let $\Sigma_g$ be a closed surface of genus $g$, let $M_g$ be its mapping class group, and let $M_g[D]$ be the subgroup generated by the $D$-th powers of Dehn twists about all simple closed curves. Let $SCC_g$ be the infinite set of simple closed curves on $\Sigma_g$, and write $T_a$ for the Dehn twist about $a$. Ivanov's presentation conjecture. For $D\geq 3$, $g\geq 4$ or $D\geq 4$, $g\in\{2,3\}$, the group $M_g[D]$ has a presentation with generators $Z_a$, corresponding to $T_a^D$ for $a\in SCC_g$, and relations

$$Z_{T_a^D(b)}=Z_a Z_b Z_a^{-1}$$

for every pair $a,b\in SCC_g$. This is a proposed presentation by braid-type relations, motivated by a question of Ivanov about relations among powers of Dehn twists.