Banach-algebra linearity criterion for holomorphic reflexivity

Let $G$ be a compactly generated Stein group. For a complex Lie group $G$, let $\operatorname{\widetilde{Lin}_{\mathbb{C}}}(G)$ denote the intersection of the kernels of all holomorphic homomorphisms from $G$ into the groups of invertible elements of Banach algebras. The algebra $\mathcal{O}(G)$ is holomorphically reflexive when it has the holomorphic reflexivity property considered in the paper.

Banach-algebra linearity conjecture. The condition

$$\operatorname{\widetilde{Lin}_{\mathbb{C}}}(G)=\{1\}$$

is necessary and sufficient for the holomorphic reflexivity of $\mathcal{O}(G)$.

The condition is already known to be necessary for Stein groups. The conjecture is proved when $G$ is abelian, discrete, or has finitely many components, but remains open for general compactly generated Stein groups with infinitely many components.