The length characterization of covers in the double-affine Bruhat order

Let $W_{\mathcal{T}}$ be the double-affine Weyl monoid, equipped with its Bruhat order $<$ and length function $\ell$. Write $x\lhd y$ when $y$ covers $x$, meaning that $x<y$ and there is no $z$ with $x<z<y$. The cover-length conjecture. For $x,y\in W_{\mathcal{T}}$, one has

$$x\lhd y \quad\Longleftrightarrow\quad x<y\text{ and }\ell(y)=\ell(x)+1.$$

The conjecture asks whether the length difference always detects covering relations in the double-affine Bruhat order, as it does for ordinary Coxeter-group Bruhat orders. The paper establishes that a cover must arise from multiplication by a reflection, but the converse characterization by length remains open in the double-affine setting.