Convex-hull characterization of the $\mathbb{F}_{2^2}$ relaxed code polytope

Let $\mathcal{E}$ be the set defined by the stated integer constraint lemma for $\mathbb{F}_{2^2}$ and by the all-ones check of length $d$. Let $\mathbb{U}_4$ be the relaxed code polytope for $\mathbb{F}_{2^2}$ associated with the all-ones checks. The notation $\operatorname{conv}(\mathcal{E})$ denotes the convex hull of $\mathcal{E}$. Convex-hull conjecture. The relaxed code polytope equals the convex hull of the integral configurations:

$$\mathbb{U}_4=\operatorname{conv}(\mathcal{E}).$$

This gives an exact polyhedral description for the all-ones check over $\mathbb{F}_{2^2}$ and would establish tightness of the relaxation in this case. The source does not state a resolution, so the conjecture is recorded as open.