The value of the minimum size for minimal uncolorable triple bi-hypergraphs of order seven

For a bi-hypergraph, a proper coloring assigns colors so that every edge contains more than one color and is not monochromatic. Let $\ell(n,r)$ denote the minimum number of edges in a minimal uncolorable $r$-uniform bi-hypergraph of order $n$.

Conjecture on $\ell(7,3)$.

$$\ell(7,3)=11.$$

The preceding results establish $10\le \ell(7,3)\le 11$. Thus the conjecture is equivalent to the nonexistence of a minimal uncolorable $3$-uniform bi-hypergraph of order $7$ with $10$ edges; the supplied text gives no resolution.