Compact-fiber conjecture for the reduced multiplicative Hitchin system

Let $G$ be a reductive group, $D$ a divisor on $\mathbb{CP}^1$, and $\omega^\vee$ the prescribed coweight data. Let $\operatorname{mHiggs}^{\mathrm{red}}_G(\mathbb{CP}^1,D,\omega^\vee)$ denote the Hamiltonian reduction of the framed multiplicative Higgs moduli space by the adjoint torus $T'$. Compact-fiber conjecture. The reduced moduli space $\operatorname{mHiggs}^{\mathrm{red}}_G(\mathbb{CP}^1,D,\omega^\vee)$ is the total space of an algebraic integrable system with generically compact smooth fibers. This is expected to resolve the non-compact-fiber problem for the multiplicative Hitchin fibration; the source gives no proof or status beyond this expectation.