Finite-line fixed-set conjecture for maximal reducible subgroups of quaternionic groups

Let $G$ be a quaternionic reflection group, or more generally a finite irreducible group of $d\times d$ matrices over the quaternions $\mathbb{H}$. A subgroup is reducible if it preserves a nontrivial proper quaternionic subspace, and it is maximal reducible if it is maximal among reducible subgroups. Finite-line fixed-set conjecture. Every maximal reducible subgroup fixes a finite number of lines.