Hamilton-cycle conjecture for involutions of type D

Let $I_n^D$ denote the involutions in the type-D signed permutation group, and let $G({\mathcal S}_n^B,X_1^D\cup X_2^D)$ be the Cayley graph with generating sets

$$X_1^D=\{t_{i,j}: i\leq j<n\},$$ $$X_2^D=\{t_{i,n}\cdot t_{j,n}:i<j<n\}.$$

Consider the restriction of this Cayley graph to its involutions, with Hamming distance measured between successive involutions. Hamilton-cycle conjecture. There is a Hamilton cycle in this restriction, with Hamming distance two, for $n\geq 4$. This is the minimal Hamming distance of any Gray code for $I_n^D$. Computational evidence motivates the conjecture, and the asserted distance is optimal; the statement remains unresolved in the supplied source.