The LQEL classification conjecture for cyclic Picard group

Let $X$ be a local quadratic entry locus manifold (LQELM), meaning a projective manifold whose general entry loci of the secant variety are quadrics, and suppose that

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

The LQEL classification conjecture. $X$ is obtained by taking linear sections and/or isomorphic projections of a rational homogeneous manifold in its natural minimal embedding.

Rational homogeneous manifolds are well understood, and the secant-defective ones are known to be LQEL manifolds and completely classified. The conjecture proposes that all LQEL manifolds with cyclic Picard group arise from these homogeneous examples by the stated operations; the source provides no resolution.