Gaiotto's scaling-limit conjecture for non-Abelian Hodge correspondence

Let $C$ be a smooth projective curve, and let $(E_0,\phi(\mathbf{q}))$ be a stable Higgs bundle on the $SL(r,\mathbb{C})$-Hitchin section. Let $(\nabla,\phi,h)$ be the corresponding solution of Hitchin's equations. For $\zeta\in\mathbb{C}^*$ and $R\in\mathbb{R}_+$, define

$$\nabla(\zeta,R):=\frac{R}{\zeta}\phi+\nabla+\zeta R\phi^\dagger.$$

Gaiotto's conjecture. The scaling limit

$$\lim_{\substack{R\to 0,\ \zeta\to 0\\ \zeta/R=\hbar}}\nabla(\zeta,R)$$

exists for every $\hbar\in\mathbb{C}^*$ and forms an $\hbar$-family of $SL(r,\mathbb{C})$-opers. This conjecture proposes a canonical construction of the correspondence between the Hitchin and de Rham moduli spaces through a scaling limit of the non-Abelian Hodge correspondence; the earlier prose formulation in the paper is a restatement of this claim.