Nagle's codegree density conjecture for $K_4^-$

For a $3$-graph $F$, let $\pi_2(F)$ denote its codegree density, namely the limit threshold for the minimum codegree required to force a copy of $F$. In particular, $K_4^-:=([4],\{123,124,134\})$ is the complete $3$-graph on four vertices with one edge removed.

Nagle's conjecture.

$$\pi_2(K_4^-)=\frac14.$$

This conjecture gives the codegree analogue of the Turán problem for the smallest non-trivial $3$-graph and is related to the characterization of $K_4^-$-free $3$-graphs through triangle-free link graphs. The source provides no resolution status, so the conjecture is recorded as open.