Existence of minimal uncolorable uniform bi-hypergraphs

Let $Bi\mathcal{H}yp(n,r)$ be the set of $r$-uniform bi-hypergraphs of order $n$, and let $\mathcal{MUC}(n,r)$ be the set of minimal uncolorable bi-hypergraphs in $Bi\mathcal{H}yp(n,r)$. For $r\ge 3$, every bi-hypergraph in $Bi\mathcal{H}yp(n,r)$ is colorable when $r\le n\le (r-1)^2$, while minimal uncolorable bi-hypergraphs are known to exist at $n=(r-1)^2+1$ in the complete case.

Existence conjecture. For any $n,r\in \mathbb{N}$ with $r\ge 3$ and $n\ge (r-1)^2+1$, the set $\mathcal{MUC}(n,r)$ is not empty.

This conjecture concerns the existence of minimal obstructions to proper coloring beyond the first possible uncolorable order and would provide a general response to the existence aspect of the problem posed by Tuza and Voloshin. The supplied text gives no resolution beyond the stated known colorability range.