Linear deformation conjecture for filiform compatible Lie algebras

Let $\mathcal{L}$ be a filiform compatible Lie algebra of dimension $n+1$.

Linear deformation conjecture. The algebra $\mathcal{L}$ is a linear deformation of the algebra $\mathcal{L}_s$ for certain $s \in \mathbb{N}^t$.

This would extend the preceding deformation results for model filiform compatible Lie algebras and reduce the study of general filiform compatible Lie algebras to linear deformations of the algebras $\mathcal{L}_s$. The source does not state whether the question is resolved.