Conjecture on Turán densities of iterated suspensions

Let $S(H)$ be the suspension of a hypergraph $H$, obtained by adjoining a new vertex and adding it to every edge of $H$, and let $S^k(H)$ denote the $k$-fold iterated suspension. Write $\pi(H)$ for the Turán density of $H$.

Iterated-suspension conjecture. For any $k\geq 2$,

$$\pi\left(S^k\left(K_2^{\{1,2\}}\right)\right)=1+\frac{1}{2^{k+1}}.$$

The preceding argument establishes the value for the first suspension example and gives the displayed lower-bound construction for general $k$; the equality claimed here is presented without a proof in the source.