Distinct Steenrod extensions of categorified quantum groups

Let $p$ be a prime, let $\mathcal{A}$ be a finite-dimensional sub-Hopf algebra of the Steenrod algebra, and let $\mathcal{U}^{c+}_{\mathcal{A}}$ be the compact analogue of $\mathcal{U}^{+}_{\mathcal{A}}$, defined using the natural Steenrod structure. Write $U^+$ for the positive half of the quantum group $\operatorname{U}_q\mathfrak{sl}(2)$, and let $\operatorname{K}_0$ denote the Grothendieck group.

$$\operatorname{K}_0(\mathcal{U}^{c+}_{\mathcal{A}}) \cong U^+ \otimes \operatorname{K}_0(\mathcal{D}(k,\mathcal{A})).$$

Distinct-extension conjecture. Any such extension decategorifies an extension of the ground field as above, and if $\mathcal{A}$ and $\mathcal{A}'$ are distinct sub-Hopf algebras, then the corresponding extensions are not equivalent as categories:

$$\mathcal{A} \not\cong \mathcal{A}' \Rightarrow \mathcal{U}_{\mathcal{A}} \not\cong \mathcal{U}_{\mathcal{A}'}.$$

This predicts that the compact categorifications retain enough information to distinguish the underlying Steenrod extensions. The supplied text does not establish the claim or give evidence resolving its status.