Monotonicity inequalities for Steinberg quotient coefficients

Let $\mathbb{X}^+$ be the dominant weights and let $\lambda,mu,mu'\inmathbb{X}^+$. Write

$$q(\lambda)=\sum_{\mu\in\mathbb{X}^+}a_{\mu}^{\lambda}s(\mu),$$

where $s(\mu)$ is the Weyl-orbit sum. Use the order relation $\uparrow$ on weights. Monotonicity conjecture. If

$$\mu-\rho\;\uparrow\;\mu'-\rho\;\uparrow\;\lambda-\rho,$$

then

$$a_{\mu}^{\lambda}\ge a_{\mu'}^{\lambda}.$$

The analogous inequalities for the coefficients $b_{\mu}^{\lambda}$ are stated as a theorem in the paper, while the corresponding inequalities for $a_{\mu}^{\lambda}$ are asserted here without a supplied resolution and remain open.