Zak's converse conjecture for Scorza varieties

Let $X \subset \mathbb{P}^N$ be an irreducible, smooth, non-degenerate, projective variety. Write $X^*$ for its projective dual, and let $\operatorname{ord}(X)$ denote its order. Say that $X$ is $k$ dual defective when the corresponding $k$-th dual defect is positive. Zak's converse conjecture. If $X$ is not $k$ dual defective for all $k < \operatorname{ord}(X)$ and

$$\deg(X^*) = \operatorname{ord}(X)+1,$$

then $X$ is a hyperquadric or a Scorza variety.

Scorza varieties are the varieties characterized by the extremal equality $\deg(X^*)=\operatorname{ord}(X)$; this conjecture proposes a converse-type classification one degree above that equality. The source formulates the conjecture in the setting where the inequality $\deg(X^*) \geq \operatorname{ord}(X)$ is known, but gives no resolution of the asserted classification.