Uniqueness failure conjecture for additive actions on degenerate hypersurfaces

Let $X\subseteq\mathbb P^{n}$ be a degenerate hypersurface, meaning that its defining homogeneous polynomial can be reduced to fewer than $n+1$ variables by a linear change of variables. Suppose that $X$ admits an induced additive action.

Additive-action multiplicity conjecture. There are at least two non-equivalent induced additive actions on $X$.

For non-degenerate hypersurfaces, induced additive actions are known to be unique up to equivalence, so this conjecture predicts that degeneracy is precisely where uniqueness fails. It is attributed in the source to Conjecture 5.19 of Arzhantsev and Zaitsev.