Uniform spectral gap conjecture for hyperbolic special groups

A group is called $C$-special or $A$-special in the sense of Haglund and Wise. For an integral chain in such a group, a spectral gap $B$ means that every integral chain not equivalent to the zero chain has stable commutator length at least $B$.

Uniform spectral gap conjecture. There is a uniform constant $B>0$ such that any hyperbolic $C$-special or $A$-special group has a spectral gap $B$ for integral chains.

This conjecture extends the uniform gap established for integral chains in hyperbolic right-angled Coxeter groups to hyperbolic special groups. The definitions of $C$-special and $A$-special groups are attributed to Haglund and Wise; the source does not state whether the conjecture has been resolved.