Regularizing adiabatic Fredholm family conjecture for the gauge-theoretic adiabatic limit

Let $Z=[-1,1]\times[0,1]$, let $\Omega$ be the target space of the displayed adiabatic family, and let $(\mathcal{F}_\epsilon:\mathcal{V}_\Gamma\to\Omega)_{\epsilon\in[0,1]}$ be equipped with the norms $\|\cdot\|^\Gamma_\epsilon$ and $\|\cdot\|^\Omega_\epsilon$ described above. A family is called regularizing when it has the regularity properties required in the definition of an adiabatic Fredholm family, and $\mathcal{C}^1$-regular when it has the stated $\mathcal{C}^1$ regularity.

Adiabatic Fredholm family conjecture. The data

$$\bigl( (\mathcal{F}_\epsilon:\mathcal{V}_\Gamma\to\Omega)_{\epsilon\in[0,1]},\|\cdot\|^\Gamma_\epsilon,\|\cdot\|^\Omega_\epsilon\bigr)$$

can be supplemented to form a regularizing $\mathcal{C}^1$-regular adiabatic Fredholm family.

This conjecture would provide the Fredholm-theoretic framework needed to make the classical adiabatic limit rigorous for the gauge-theoretic equations under discussion. The supplied passage does not state whether it has been resolved.