Splitness conjecture for super-modular categories with modular quotient S-matrix

Let $\mathcal{B}$ be a super-modular category, and let $\hat{S}$ be the $S$-matrix of a modular category $\mathcal{D}$. Suppose that the $S$-matrix of $\mathcal{B}$ is

$$\hat{S}\otimes\begin{pmatrix}1&1\\1&1\end{pmatrix}.$$

Splitness conjecture. Then $\mathcal{B}$ is split super-modular.

Here, split super-modular means that $\mathcal{B}$ is equivalent to a Deligne product $\mathcal{D}'\boxtimes\sVec$ for some modular category $\mathcal{D}'$. The claim is motivated by the classification evidence available in the paper up to rank $6$; the source does not state a proof or a resolution.