Conjecture on linearly normal Legendrian varieties

Let $V$ be a vector space and let $X\subset {\mathbb P}(V)$ be a smooth linearly normal Legendrian variety, where linearly normal means that the embedding is given by a complete linear system. A subadjoint variety is one of the standard Legendrian varieties associated with adjoint varieties.

Conjecture on linearly normal Legendrian varieties. $X$ is one of the subadjoint varieties.

This conjecture is posed as an open classification problem for smooth linearly normal Legendrian varieties. The source gives no resolution or partial classification sufficient to determine its status more precisely.