Smooth splitting conjecture for the twisted bundle over Hitchin's moduli space

Let $V_z$ be the twisted bundle obtained as the globally defined direct sum of the two locally defined bundles $V_z^1$ and $V_z^2$ over the relevant cover of the Hitchin moduli space. Smooth splitting conjecture. The twisted bundle $V_z$ admits a $C^{\infty}$ splitting

$$V_z=V_z^1\oplus V_z^2.$$

Such a splitting would globalize the two sheet-dependent summands and is needed to construct the corresponding hyperholomorphic connection; the source gives no resolution.