The generalized cubic-scroll description conjecture for a cubic Hodge locus

Let $V_{[Z]}$ be the Hodge locus associated with $Z=r\mathbb{P}^{\frac n2}+\check r\check{\mathbb{P}}^{\frac n2}$, where the two linear cycles meet in $\mathbb{P}^m$. Generalized cubic-scroll conjecture. If $(d,m)=(3,\frac n2-2)$ and $(r,\check r)=(1,-1)$, then $V_{[Z]}$ is smooth and parameterizes hypersurfaces containing a generalized cubic scroll. The supplied text gives the geometric description and points to cited definitions, but provides no resolution of this conjectural statement.