Dimension bound for solvable compatible extensions of $\mathcal{L}_s$

Let $\mathcal{L}_s$ be the filiform compatible Lie algebra considered in the paper, and let $\underline{n_s}$ denote the parameter used for its dimension. A solvable compatible Lie extension of $\mathcal{L}_s$ is a solvable compatible Lie algebra having $\mathcal{L}_s$ as its nilradical.

Dimension-bound conjecture. There are no solvable compatible Lie extensions of $\mathcal{L}_s$ of dimension greater than $\underline{n_s}+5$.

The question is motivated by the construction of solvable compatible Lie extensions of dimensions up to $\underline{n_t}+1+k$ for $1\leq k\leq 4$. The paper closes by presenting this as an open question, so the asserted upper bound remains open.