Parameter-scaling conjecture for locally free representations

Let $H(k)=H(C,kD,\Omega)$ for $k\geq 2$, and let $\mathbf{r}$ be a rank vector. Define the number of parameters by

$$\mu_{H(k)}(\mathbf{r})=\dim \operatorname{rep}_{\rm l.f.}(H(k),\mathbf{r})-\max\\{\dim \mathcal{O}_M\mid M\in\operatorname{rep}_{\rm l.f.}(H(k),\mathbf{r})\\}.$$

Parameter-scaling conjecture. For all rank vectors $\mathbf{r}$,

$$\mu_{H(k)}(\mathbf{r})=k\cdot\mu_{H(1)}(\mathbf{r}).$$

This would describe how the moduli-theoretic parameter count changes under scaling the symmetrizer; the paper does not establish the assertion in general.