On the symplectic forms of Groechenig's Higgs moduli over an elliptic curve

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types $\text{wt}{D}_4$, $\text{wt}{E}_6$, $\text{wt}{E}_7$, and $\text{wt}{E}_8$, for the $\mathrm{\Gamma }$-Hilbert schemes of $T^{\ast }E$ for cyclic groups $\mathrm{\Gamma }$ with $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }\in \{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.