Parity conjecture for maximum equal-parameter town families

Let $m_{k,n}(a,a)$ be the maximum size of a family of subsets of $[n]$ whose intersections are all congruent to $a$ modulo $k$, with $n\geq k\geq 2$. Parity conjecture. If $n\geq k+a$, then $m_{k,n}(a,a)$ is even.

The conjecture concerns the parity of the extremal size rather than its exact value, which the source describes as difficult to determine in general. It is based on computational evidence for small parameters and remains open.