Deodhar's inequality for the double-affine Bruhat order

Let $W_{\mathcal{T}}$ be the double-affine Weyl monoid, let $\widetilde{\Delta}^+$ denote the positive double-affine real roots, and let $\ell$ be the length function. Deodhar's inequality. If $x,y,z\in W_{\mathcal{T}}$ satisfy $x\leq y\leq z$, then

$$\#\{\beta[n]\in\widetilde{\Delta}^+\mid x\leq ys_{\beta[n]}\leq z\}\geq\ell(z)-\ell(x).$$

In finite and single-affine cases, the analogous inequality was conjectured by Deodhar and has since been proved by many authors. Its validity in the double-affine setting is the unresolved assertion made here.