Veselinov–Marinov linear bound conjecture for distinct-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 linear bound conjecture. If $a\neq b$, then

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

For prime moduli, the source records this bound as known in the corresponding general upper-bound result, but presents the stated all-moduli assertion as a conjecture; the general question remains open for $k\geq 3$.