Nilpotent one-parameter subgroup conjecture for Legendrian subvarieties

Let $V$ be a symplectic vector space, let $X'\subset \P(V)$ be an irreducible Legendrian subvariety, let $g\in\mathfrak{sp}(V)$ be a nilpotent endomorphism, and let $m$ be an integer such that

$$g^m\ne 0,\qquad g^{m+1}=0.$$

Assume that the action of $\exp(tg)$ preserves $X'$, and that $X'$ is singular at points of the image of the rational map $g^m(X')$. Nilpotent one-parameter subgroup conjecture. Then $X'$ is decomposable.

This is presented as a special case to which the automorphism conjecture is reduced, and is not covered by the paper's main evidence theorem. Its resolution is not given in the supplied text.