The monotonicity conjecture for diagonal $(m,m)$-town families modulo $k$

Let $[n]=\{1,2,\ldots,n\}$, let $\mathscr{P}(n)$ be the power set of $[n]$, and let a family be an $(a,b)$-town modulo $k$ when all its sets have cardinality congruent to $a$ modulo $k$ and every pair of distinct sets has intersection cardinality congruent to $b$ modulo $k$. For $i=1,2$, let $\mathcal{F}_i\subseteq\mathscr{P}(n)$ be a $(m_i,m_i)$-town modulo $k$ family of maximum size, where $0\leq m_1,m_2\leq k-1$. Monotonicity conjecture. If $m_1<m_2$, then

$$|\mathcal{F}_1|\geq |\mathcal{F}_2|.$$

This conjecture arises from numerical evidence and concerns the relative sizes of diagonal town families when the modulus $k$ need not be prime. The linear upper bound $n$ is already known when the two town parameters differ modulo some prime divisor of $k$; the remaining cases, particularly those with $a\equiv b$ modulo every prime divisor of $k$, are not fully understood.