Veselinov–Marinov monotonicity conjecture for equal-parameter town families

Let $m_{k,n}(a,b)$ denote the maximum size of a family of subsets of $[n]$ whose pairwise intersections are congruent to $a$ modulo $k$ when the two subsets coincide and to $b$ modulo $k$ otherwise. Let $a,b,k,n$ be integers with $n\geq k\geq 2$ and $0\leq a,b\leq k-1$. Veselinov–Marinov's monotonicity conjecture. If $a<b$, then

$$m_{k,n}(a,a)\geq m_{k,n}(b,b).$$

This conjecture compares the maximum sizes of equal-parameter town families as the prescribed self-intersection residue increases; it is part of the largely open problem of determining $m_{k,n}(a,b)$ for moduli $k\geq 3$.