Conjecture on definability and model companions for Γ-commutative operator fields

Let $\mathcal D$ be an operator structure with a specified $\Gamma$-commutativity condition, and let $\mathcal L_{\underline{\mathcal D}}(F)$ be the corresponding language. For a $\mathcal D^\Gamma$-field $(K,e)$, consider the set

$$\{a\in K: a \text{ has a $p$-th root in a $\mathcal D$-field extension of } K\}.$$

Definability and model-companion conjecture. (1) There is an $\mathcal L_{\underline{\mathcal D}}(F)$-formula which uniformly describes this set in any $\underline{\mathcal D}^\Gamma$-field. (2) The model companion of $\underline{\mathcal D}^\Gamma$-fields exists even when Assumption~ is dropped.

The conjecture concerns whether $\Gamma$-commutativity restores first-order definability of having a $p$-th root in an operator-field extension, and consequently permits existence of the model companion without the stated assumption. The text gives an example with pairwise commuting operators where the relevant partial type is equivalent to a single formula, but does not establish either assertion in general.