The higher-girth face-number conjecture for simplicial complexes

Let $\Gamma$ be a $(d-1)$-dimensional simplicial complex with $n$ vertices. Let $f_{i-1}(\Gamma)$ denote the number of $(i-1)$-dimensional faces, and let $\operatorname{gr}_{p-1}(\Gamma)$ denote its $(p-1)$-girth. For some integers $p,r\geq 2$, assume that $\operatorname{gr}_{p-1}(\Gamma)>2p+2r-4$. Let $a_{p,r,i}$ be the recursively defined exponents

$$a_{2,r,i}=1+1/r+\ldots+1/r^{i-1},$$ $$a_{p,r,p-1}=p-1,$$ $$a_{p,r,i}=\frac{1}{2}a_{p-1,r,i-1}+\frac{1}{2}a_{p,r,i-1}+1\quad (p\geq 3, i\geq p).$$

The higher-girth face-number conjecture. There is a constant $C_{p,r,i}$ depending only on $p,r,i$ such that

$$f_{i-1}(\Gamma)\leq C_{p,r,i}d^{i-a_{p,r,i}}n^{a_{p,r,i}}.$$

The conjecture seeks upper bounds for face numbers from the dimension, number of vertices, and $(p-1)$-girth. The recursively defined exponents increase with the relevant parameters and approach $2p-3+1/(r-1)$ as $i$ tends to infinity; the asserted bound is not established in the supplied text.