Haiman has constructed in the paper the unusual tautological bundle $P$, called Procesi bundle, of rank $n!$ over the Hilbert schemes of points on the affine plane in the following way.
Let $H _ { n } = \mathrm { Hilb } ^ { n } \left( \mathbb { C } ^ { 2 } \right)$ be the Hilbert scheme of $n$-points of the affine plane $\mathbb { C } ^ { 2 }$, and let $S ^ { n } \mathbb { C } ^ { 2 }$ be the $n$-th symmetric product of $\mathbb { C } ^ { 2 }$. We define $X_n$ as a fiber product via $$ \begin{array} {ccc} X _ { n } &\stackrel { f } \longrightarrow & \mathbb { C } ^ { 2 n } \\ \rho \downarrow & & \downarrow \\ H _ { n }& \stackrel { \sigma } { \longrightarrow } & S ^ { n } \mathbb { C } ^ { 2 } \end{array} $$ where $\sigma$ is the Hibert-Chow map. The Procesi bundle over the Hilbert scheme $H_n$ is defined by the pushforward $P = \rho _ { * } \mathcal { O } _ { X _ { n } }$ of the structure sheaf of $X_n$.
Question: is the Procesi bundle equipped with a hyperholomorphic connection over the hyper-K\"ahler manifold $H_n$?
