Virtual abelianity of the quotient by the tree-lattice commensurator’s monolith

Determine whether the quotient C_d/M_d is virtually abelian, where C_d is the commensurator of the standard cocompact lattice in Aut(T_d) and M_d is the monolith of C_d.

Background

The paper constructs an infinite-index normal subgroup N_d of C_d such that C_d/N_d is free abelian of infinite rank, and knows that the monolith M_d is contained in N_d. It does not know whether M_d=N_d, nor does it otherwise fully describe the quotient C_d/M_d. The first explicit question asks whether this quotient is at least virtually abelian.