Vatne's conjecture on Cohen–Macaulay triple structures

Let $L \subset \mathbb{P}^N$ be a subspace of dimension $n \geq 6$ and arbitrary codimension. Let $L^{(2)}$ denote the second infinitesimal neighborhood of $L$, and let $Y \subset \mathbb{P}^N$ be a Cohen–Macaulay subscheme of degree three satisfying

$$L \subset Y \subset L^{(2)}.$$

Vatne's conjecture. There is a degree-two Cohen–Macaulay subscheme $Z$ such that

$$L \subset Z \subset Y.$$

This conjecture is proposed as a test toward classifying Cohen–Macaulay multiple structures on a linear space up to degree three. Its status is not resolved in the supplied source.