The lower-bound sharpness conjecture for $K_5^<$

Let $K_5^<$ be the $3$-graph on five vertices with eight edges whose two missing edges intersect in exactly two vertices. The supplied bounds are

$$\frac{5}{9}\leq\pi(K_5^<),\quad \frac{1}{3}\leq\sigma(K_5^<),\quad \frac{1}{2}\leq\pi_2(K_5^<),\quad\text{and}\quad \frac{1}{2}\leq\pi_u(K_5^<).$$

The lower-bound sharpness conjecture for $K_5^<$. All four lower bounds are sharp:

$$\pi(K_5^<)=\frac{5}{9},\qquad \sigma(K_5^<)=\frac{1}{3},\qquad \pi_2(K_5^<)=\frac{1}{2},\qquad\text{and}\qquad \pi_u(K_5^<)=\frac{1}{2}.$$

The text gives flag-algebra upper bounds that are strictly above these lower bounds, so the conjecture remains open in the supplied material.