Conjecture on leading matrices of distinguished involutions

Let $\tilde{d538}_n$ be the extended affine Weyl group of type $\tilde{d538}_n$, let $d740$ be a two-sided cell indexed by a partition $d706\vdash n+1$, and let $M_{d706}$ be the associated module with basis $d401_{d706}$. Write $d_1,\ldots,d_{N_{d706}}$ for the distinguished involutions in the relevant left cell decomposition, and let $E_{ii}$ denote the elementary matrix with a $1$ in position $(i,i)$ and zero elsewhere.

Leading-matrix conjecture. There exists an ordering of the basis $d401_{d706}$ of $M_{d706}$ such that the leading matrices of the distinguished involutions satisfy

$$\mathfrak{c}_{\lambda}(d_i)=E_{ii}$$

for $1\leq i\leq N_{d706}$.

The claim extends the established property of the distinguished involution $d700_{d706'}$, whose leading matrix is $E_{d758_{d706},d758_{d706}}$. It concerns the matrix realization of Lusztig's asymptotic algebra and remains conjectural in the source.