Extensibility conjecture for doubly-periodic anti-self-dual instantons

Let $A$ be an anti-self-dual connection on a bundle $E\rightarrow T\times\mathbb{C}$, with curvature satisfying $|F_A|\sim O(|w|^{-2})$. A holomorphic structure $(E,\overline{\partial}_A)$ is induced by $A$. Extensibility conjecture. If

$$|F_A|\sim O(|w|^{-2}),$$

then there is a holomorphic vector bundle ${\cal E}\rightarrow T\times\mathbb{P}^1$ such that

$${\cal E}|_{T\times(\mathbb{P}^1\setminus\{\infty\})}\simeq(E,\overline{\partial}_A).$$

In other words, $A$ is extensible. The paper presents this as an expected analytical result: it would show that the stated curvature decay is sufficient for extensibility. Whether this implication holds is left open.