The covering-number characterization of S-GQPQs

Let $(k)^3$ denote the set of triples under consideration, let $S$ be a subset of $(k)$, and let $H\subseteq (k)^3$. A set of triples has lines as hypergraph edges, and its covering number $\rho(H)$ is the minimum number of lines whose union covers it.

The covering-number converse conjecture. If

$$\rho\bigl(H\cap (S\times (k))\bigr)=|S|,$$

then $H$ contains an $S$-GQPQ.

This is proposed as the converse to the preceding necessary condition, giving a characterization of the existence of an $S$-GQPQ by the covering number. The supplied text gives no resolution, so the conjecture remains open.