Exponential relation between maximal-clique deficiency and layered-tree parameter

Let $f(n,k)$ denote the maximal-clique deficiency parameter and let $c(n,k)$ denote the layered-tree parameter for $k$-uniform hypergraphs, as defined in the paper. Exponential relation conjecture. For every fixed integer $k\ge3$,

$$f(n,k)=\Theta_k\!\left(2^{c(n,k)}\right)$$

as $n\to\infty$. Equivalently,

$$\log f(n,k)=c(n,k)+O_k(1).$$

Here the constants may depend on $k$. The preceding results establish this relation for $k=3$ and show that the two parameters have substantially different growth in that case; the conjecture proposes the corresponding exponential relationship for every fixed uniformity $k\ge3$.