Nonexistence of pairs of maximal frequency squares of half frequency

A frequency square of type $(n;\lambda)$ is an $n\times n$ array in which each symbol occurs exactly $\lambda$ times in every row and every column. Two frequency squares are orthogonal if each ordered pair of symbols occurs equally often in corresponding cells. A set of $k$ mutually orthogonal frequency squares of type $(n;\lambda)$ is called $k$-MOFS$(n;\lambda)$, and it is maximal if no further frequency square of type $(n;\lambda)$ is orthogonal to every member of the set; such a set is denoted $k$-maxMOFS$(n;\lambda)$. Nonexistence conjecture for pairs of maximal frequency squares. If $n$ is even, then there does not exist a set of $2$-maxMOFS$(n;n/2)$. The conjecture is known when $4$ divides $n$, and it holds for $n<8$ by the stated nonexistence results. The case in which $n/2$ is odd remains open.