All plausible tau-parities are eventually actual

Let $k$ be fixed. For an orthogonal array of strength two, $\tau$-parity records the parity data associated with its three-way choices of distinct coordinates, and a $\tau$-parity is plausible if it satisfies all restrictions established in the paper. A plausible $\tau$-parity is actual for $(k,N)$ if it is realised by some $\mathrm{OA}(k,N)$.

Actuality conjecture. For any fixed $k$ and all sufficiently large $N$, all plausible $\tau$-parities are actual for $(k,N)$.

The conjecture asserts that, once the number of columns is fixed and the order is sufficiently large, the known parity restrictions are the only restrictions. The preceding monotonicity theorem shows that actuality at one order persists for all sufficiently large orders, but does not establish that every plausible parity is realised.