Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

Let $(X,D)$ and $(X^{\mathrm{\prime }},D^{\mathrm{\prime }})$ be two compact Riemann surfaces of genus $g\ge 4$ with the set of marked points $D\subset X$ and $D^{\mathrm{\prime }}\subset X^{\mathrm{\prime }}$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let ${\mathcal{N}}_{\text{Sp}}(2m,\alpha ,L)$ and ${\mathcal{N}}_{\text{Sp}}^{\mathrm{\prime }}(2m,\alpha ,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X^{\mathrm{\prime }}$ respectively, with rank $2m$ and fixed parabolic structure $\alpha$, with the symplectic form taking values in $L$. We prove that if ${\mathcal{N}}_{\text{Sp}}(2m,\alpha ,L)$ is isomorphic to ${\mathcal{N}}_{\text{Sp}}^{\mathrm{\prime }}(2m,\alpha ,L)$, then there exist an isomorphism between $X$ and $X^{\mathrm{\prime }}$ sending $D$ to $D^{\mathrm{\prime }}$.