The dual-defective-to-LQEL conjecture

Let $X$ be a dual defective manifold, meaning that its dual variety is not a hypersurface, and suppose that

$$\operatorname{Pic}(X)\cong\mathbb{Z}\langle H\rangle.$$

The dual-defective-to-LQEL conjecture. $X$ is a local quadratic entry locus manifold (LQELM).

This conjecture would identify dual defective manifolds with cyclic Picard group as secant defective varieties of a particularly rigid, locally quadratic type. The paper gives evidence for the claim, but the source provides no resolution.