Existence of infinite cubulable groups with fixed-point property $(\mathrm{FW}_n)$

A group is cubulable if it acts properly and cocompactly on a median graph. For $n\geq 1$, $(\mathrm{FW}_n)$ denotes the fixed-point property for actions on median graphs of cubical dimension at most $n$. Existence conjecture. For every $n\geq 1$, there exists an infinite cubulable group satisfying $(\mathrm{FW}_n)$ with no proper finite-index subgroup. The surrounding discussion gives constructions of cubulable groups with fixed-point properties and explains why controlling finite-index subgroups is difficult; the existence assertion is left as an open problem.