The subgroup membership conjecture for the groups $B(P_{3,k})$

For each integer $k\geq 2$, let

$$B(P_{3,k})=\operatorname{Gp}\langle a,b,c\mid [a,b^k]=1,[a,c]=1\rangle.$$

The subgroup membership conjecture for $B(P_{3,k})$. The subgroup membership problem is decidable in $B(P_{3,k})$ for every $k\geq 2$. The subgroup membership problem is known to be undecidable for related groups such as $B(P_{3,k})$'s submonoid membership problem when $k\geq 2$, while decidability of subgroup membership here is presented as an open problem motivated by the absence of an embedded subgroup isomorphic to $F_2\times F_2$.