The minimum-size conjecture for minimal uncolorable uniform bi-hypergraphs

Let $m(r)$ be the smallest positive integer for which there exists a minimal uncolorable $r$-uniform bi-hypergraph with $m(r)$ edges. Complete $r$-uniform bi-hypergraphs on $(r-1)^2+1$ vertices have $\binom{(r-1)^2+1}{r}$ edges.

Minimum-size conjecture. For any $r\in \mathbb{N}$ with $r\ge 4$,

$$m(r)=\binom{(r-1)^2+1}{r}.$$

The conjecture predicts that the complete construction gives the smallest possible minimal uncolorable $r$-uniform bi-hypergraph for every $r\ge 4$. For $r=4$, the source reduces this to proving colorability of every $4$-uniform bi-hypergraph of order at most $49$ and size at most $209$; the general conjecture remains unresolved in the supplied text.