Extremal lower bound for the suspended 6-cycle

Let $C_6$ be the $6$-cycle, let $C_6(t)$ denote the associated $3$-uniform hypergraph obtained by adjoining a new vertex to every edge of $C_6$, and let $\operatorname{ex}(n,H)$ denote the $n$-vertex extremal number of $H$. Extremal lower-bound conjecture for $C_6(t)$. There is a constant $C>0$ such that, for all $t\geq C$,

$$\operatorname{ex}(n,C_6(t))\geq\Omega\left(n^{17/6}\right).$$

The paper identifies this as one of the smallest interesting unknown cases of the preceding KST-type conjecture; the matching upper bound is known from its main extremal theorem because $C_6$ is Sidorenko.