Häggkvist's avoidability conjecture for $(m,m,m)$-arrays

An $(m,m,m)$-array is an $n\times n$ array of sets of size at most $m$, such that each number in $[n]$ occurs at most $m$ times among the sets in any row or column. It is avoidable if there exists an $n\times n$ Latin square $L$ such that no entry of $L$ is contained in the set in the corresponding cell of the array. Häggkvist's avoidability conjecture. There exists a constant $\gamma>0$ such that if $m<\gamma n$, then every $(m,m,m)$-array is avoidable. This conjecture asks for a uniform linear range in which all such arrays admit an avoiding Latin square; the source gives no resolution here.