Minimum-order conjecture for Maker-win hypergraphs with Ophelia draws

A hypergraph is a pair consisting of a vertex set and a collection of subsets of that vertex set, called hyperedges. In the Tic-Tac-Toe game on a hypergraph, Maker wins if Maker can occupy a hyperedge, while Ophelia can force a draw if she can prevent either player from occupying a hyperedge. Minimum-order conjecture. Any hypergraph that is Maker win and where Ophelia can force a draw must have at least $20$ vertices. The conjecture is motivated by the verified example on the transversal design $\mathrm{TD}(4,5)$, which has $20$ vertices and is both Maker win and Ophelia draw; the preceding discussion reports no smaller example, but does not establish the lower bound.