Automorphism conjecture for indecomposable Legendrian subvarieties

Let $X\subset \P^{2n-1}$ be an irreducible indecomposable Legendrian subvariety, and let $G<\P\operatorname{GL}_{2n}$ be a connected subgroup preserving $X$. Automorphism conjecture. Then $G$ is contained in the image of the natural map

$$\operatorname{Sp}_{2n}\longrightarrow \P\operatorname{GL}_{2n}.$$

This conjecture asserts that connected projective automorphisms of an irreducible indecomposable Legendrian subvariety preserve the ambient contact structure. The paper proves it under minor assumptions, including in the smooth case, but the general singular case remains open.