Cuspidal short exact sequence conjecture in non-simply laced affine type

Let $\Phi_+$ be the set of positive roots, and let $\rho\in\Phi_+^{\mathrm{re}}$. A pair $(\beta,\gamma)$ is a real minimal pair for $\rho$ when it is a minimal pair in the sense used for the root decomposition of $\rho$. Write $L_\rho$ for the cuspidal module associated with $\rho$, and let $L(\beta,\gamma)$ denote the other module occurring in the induction product $L_\beta\circ L_\gamma$. For $p_{\beta,\gamma}$ as defined in the preceding discussion, the known sequence in the symmetric case is

$$0 \longrightarrow L_\rho\langle p_{\beta,\gamma}-(\beta,\gamma)\rangle \longrightarrow L_\beta\circ L_\gamma \longrightarrow L(\beta,\gamma) \longrightarrow 0.$$

Cuspidal short exact sequence conjecture. For non-symmetric $\mathsf{C}$, let $\rho\in\Phi_+^{\mathrm{re}}$, and let $(\beta,\gamma)$ be a real minimal pair for $\rho$. Then there still is a short exact sequence of the form

$$0 \longrightarrow L_\rho\langle p_{\beta,\gamma}-(\beta,\gamma)\rangle \longrightarrow L_\beta\circ L_\gamma \longrightarrow L(\beta,\gamma) \longrightarrow 0.$$

This proposes that the symmetric-type short exact sequence for cuspidal modules persists in non-simply laced affine types. The source notes that an analogous result is known for all finite types, but does not provide a proof or resolution for the affine non-symmetric case.