Conjecture on the largest prime divisor of rank-proportion denominators

Let $c_k$ be the limiting proportion of vertices of rank $k$, and let $denom(c_k)$ denote its denominator when $c_k$ is written in lowest terms. Denominator prime-divisor bound conjecture. The largest prime divisor of $denom(c_k)$ is at most as large as some relatively slowly growing function of $k$, possibly $2^{k+1}+1$. The authors motivate this conjecture with the observed factorizations for $k\leq 5$, but the precise slowly growing function is not specified.