The index invariance conjecture for isomorphic open subgroups

Let $G$ be a torsion-free just-infinite $p$-adic analytic pro-$p$ group, and let $H,K$ be open subgroups of $G$.

Index invariance conjecture. If $H\simeq K$, then

$$[G:H]=[G:K].$$

This is the group version of the preceding conjecture for Lie lattices and is motivated by the analogous three-dimensional result. The supplied text does not indicate whether the conjecture has been resolved.