Stein manifolds are complex analytic submanifolds of some $\mathbb{ C}^N.$

(A version of) Grauert's theorem states that on a Stein manifold $X$ every continuous map $g\colon X\to G$ to a complex Lie group $G$ is homotopic to a holomorphic map, see Gromov "Oka's principle for holomorphic sections of elliptic bundles".

This theorem was generalized to the case where $G$ is an infinite dimensional complex Banach Lie group, see Bungart "On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers".

I would like to know if the theorem still holds when $G$ is replaced by an infinite dimensional complex Frechet Lie group or at least in the case of $G=C^\infty(M,\mathbb{S}L(2,\mathbb{C}))$ for a compact mannifold $M.$