The codegree-squared density conjecture for $K_5^4$

Let $K_5^4$ be the $4$-uniform hypergraph on five vertices with five edges. The matrix construction described in the text gives

$$\frac{31}{64}\leq\sigma(K_5^4)\leq 0.55241250.$$

The codegree-squared density conjecture for $K_5^4$. The lower bound is tight:

$$\sigma(K_5^4)=\frac{31}{64}.$$

The lower bound comes from the construction and the upper bound from flag algebras; the supplied text does not resolve whether equality holds.