Conjecture on limiting 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 $S^t(H)$ denote the $t$-fold iterated suspension. Let $R(H)$ be the object associated with $H$ in the paper's definition of the limiting construction, and let $\pi(H)$ denote its Turán density.

Iterated-suspension limit conjecture. For all hypergraphs $H$,

$$\lim_{t\to\infty}\pi\left(S^t(H)\right)=|R(H)|-1.$$

The source proves the monotonic inequality $\pi(S(H))\leq\pi(H)$ and notes preservation of degeneracy under suspension, but does not establish this limiting formula.