Lee's conjecture that the fractional Steiner triple system threshold is $2/3$

Let $\theta_{\operatorname{STS}}$ denote the minimum codegree threshold for Steiner triple systems, and let $\theta_{\operatorname{STS}}^*$ denote the corresponding fractional threshold. Lee's conjecture.

$$\theta_{\operatorname{STS}}^*=\frac{2}{3}.$$

The paper notes that this value would be strictly smaller than the ordinary threshold and would imply a substantial strengthening of the known bounds; it remains open.