Finiteness of the exponential generating limit for pattern-occurrence polynomials

For a pattern $\xi\in S_k$, let $c_n^\xi(x)$ denote the pattern-occurrence generating polynomial over $S_n$, so that its coefficients enumerate permutations according to the number of occurrences of $\xi$. Finiteness conjecture. For every pattern $\xi\in\bigcup_k S_k$ and every $x\in(0,1)$, the limit

$$\lim_{n\to\infty}\bigl(c_n^\xi(x)\bigr)^{1/n}$$

is finite. The source establishes existence of this limit by subadditivity, but does not establish the asserted finiteness in the stated generality.