Realization conjecture for the reverse-braided module category of a vertex operator algebra

Let $V$ be a rational, $C_2$-cofinite vertex operator algebra, and let $coverline{\mathcal{C}_V}$ denote its modular tensor category equipped with the reverse braiding. Realization conjecture. There is a rational, $C_2$-cofinite vertex operator algebra $U$ such that

$$\overline{\mathcal{C}_V}\simeq\mathcal{C}_U$$

as braided categories. This asks whether reversing the braiding of a modular tensor category arising from a rational vertex operator algebra can again be realized by such an algebra; the source presents it as a natural question and gives no resolution.