The Gelfand-transform criterion for elementary generation in Banach algebras

Let $R$ be a commutative, semisimple, unital complex Banach algebra with maximal ideal space $X_R$ equipped with the weak-$\ast$ topology induced from $R^*:=\mathcal{L}(R;\mathbb{C})$. Let $\widehat{\cdot}:R\to C(X_R;\mathbb{C})$ be the Gelfand transform. For $F\in SL_n(R)$, let $\widehat F$ be the matrix obtained by applying the Gelfand transform to its entries, and let $\pi(\widehat F)$ denote the homotopy class of the map $\varphi\mapsto\widehat F(\varphi):X_R\to SL_n(\mathbb{C})$. Write $E_n(R)$ for the subgroup of $SL_n(R)$ generated by elementary matrices.

Gelfand-transform criterion. $F\in SL_n(R)$ belongs to $E_n(R)$ if and only if

$$\pi(\widehat F)=0.$$

This asks whether elementary generation over a commutative semisimple Banach algebra is characterized exactly by the null-homotopy of the associated Gelfand-transform map. The supplied text presents it as a question/conjecture; no resolution is given here.