Linear upper bound conjecture for the longest solvable board

For an odd integer $n\geq 3$, let $ML(n)$ be the maximum length of a solvable board of size $n$:

$$ML(n):=\max_{A\in Sol_n}\{\operatorname{length} A\}.$$

Linear upper bound conjecture. As $n\to\infty$, $ML(n)=O(n)$.

The construction preceding this conjecture gives solvable boards of length $3n$ for all odd $n\geq 11$, while the trivial general upper bound is $ML(n)\leq n^2$. The conjecture asserts that the maximal length grows only linearly with the board size.