Twisted cotangent description of the Mukai-dual section space

Let $M$ be a compact hyperkähler manifold, let ${\cal S}$ be a connected component of the moduli of autodual connections on a complex vector bundle ${\cal B}$, and assume that ${\cal S}$ contains a Hermitian autodual point $B$. Let $\widehat M$ be the moduli space of Hermitian autodual connections on ${\cal B}$, and assume that the connected component of $\widehat M$ containing $B$ is smooth and compact. Then ${\cal S}$ is identified with the space $Sec(\widehat M)$ of twistor lines in $\operatorname{Tw}(\widehat M)$. If $\omega\in H^1(\Omega^1M)$ is the Kähler class, let $\Omega_\omega M$ denote the associated twisted cotangent bundle, defined by the extension determined by $\omega$.

Twisted cotangent conjecture. Under these assumptions, there exists a natural isomorphism of complex manifolds

$$Sec(\widehat M)\cong\Omega_\omega M.$$

The conjecture gives a geometric description of the autodual-connection moduli component in terms of a twisted cotangent bundle. The text presents it as conditional on an affirmative answer to a stronger form of a preceding question, and gives no resolution status.