Grading of affine Weyl semi-groups of Kac-Moody type

For any Kac-Moody root data $\mathcal{D}$, D. Muthiah and D. Orr have defined a partial order on the semi-direct product $W^{+}$ of the integral Tits cone with the vectorial Weyl group of $\mathcal{D}$, and a strictly compatible $\mathbb{Z}$-valued length function. We classify covers for this order and show that this length function defines a $\mathbb{Z}$-grading of $W^{+}$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.