The linear covering-set conjecture for generalized shuffle squares

Fix an alphabet of size $k$. Let $E_n$ be the set of all even $k$-ary words of length $n$, and let $S_n$ be the set of all permutations of $[n]$. Define $m_k(n)$ to be the minimum size of a set of permutations whose associated shuffle squares cover every word in $E_n$. The linear covering-set conjecture. For every $k\in\mathbb{N}$ there exists a constant $c_k$ such that

$$m_k(n)\leqslant c_kn$$

for all $n\geqslant 1$. The known binary bound is linear, and the conjecture asks for an analogous linear bound for every fixed alphabet size; it remains open.