Conjecture on Property FW for irreducible lattices in higher-rank semisimple Lie groups

Let $S$ be a connected semisimple Lie group with no compact simple factor, whose Lie algebra has real rank at least $2$. An irreducible lattice is a lattice in $S$ satisfying the usual irreducibility condition. Property FW conjecture. Every irreducible lattice in $S$ has Property FW.

Property FW is a fixed-point property for actions on spaces with walls. The conjecture proposes that higher-rank irreducible lattices, including lattices with the Haagerup Property, nevertheless satisfy this stronger fixed-point property; its status is not resolved in the supplied source.