Independent-neighbourhoods conjecture for 3-graph Turán density

Let $\pi(\operatorname{odd\ cycle\ in\ link\ graph},F_{3,2})$ denote the maximum asymptotic edge density of a 3-graph containing no copy of $F_{3,2}$ and having no odd cycle in any link graph. Independent-neighbourhoods conjecture.

$$\pi(\operatorname{odd\ cycle\ in\ link\ graph},F_{3,2})=1/4,$$

with the stable extremal configuration given by Construction~. The source says this version should be stable and essentially uniquely attained by that construction, but gives no resolution.