Surjectivity conjecture for the weakly integral Witt group

Let $\mathcal{W}$ be the Witt group of non-degenerate braided fusion categories, let $s\mathcal{W}$ be the corresponding group of slightly degenerate braided fusion categories, and let $\mathcal{W}_{\mathbb{Q}}$ and $s\mathcal{W}_{\mathbb{Q}}$ denote their weakly integral Witt subgroups. Let $S:\mathcal{W}\to s\mathcal{W}$ be the group homomorphism whose kernel is $\mathcal{W}_{Ising}\cong\mathbb{Z}/16\mathbb{Z}$. Weakly integral Witt-group conjecture. The restriction

$$\left.S\right|_{\mathcal{W}_{\mathbb{Q}}}:\mathcal{W}_{\mathbb{Q}}\to s\mathcal{W}_{\mathbb{Q}}$$

is surjective, equivalently,

$$\mathcal{W}_{\mathbb{Q}}\cong s\mathcal{W}_{\mathbb{Q}}\times\mathcal{W}_{Ising}.$$

This is the weakly integral truncation of the question of whether $S$ is surjective, which is related to the minimal modular extension conjecture for slightly degenerate braided fusion categories. The source states that the general surjectivity question is unresolved, and proposes surjectivity on weakly integral Witt classes.