The asymptotic unlabeled compression conjecture for powers of $C_5$

Let $C_5$ be the family defined earlier in the source, let $\mathcal F^{*n}$ denote its $n$-fold join, and let $\mathrm{UCS}(\mathcal F)$ be the minimum size of an unlabeled compression scheme for $\mathcal F$. The $C_5$ asymptotic conjecture.

$$\lim_{n\to\infty}\frac{\mathrm{UCS}(C_5^{*n})}{n}$$

exists and is strictly larger than $2$. The question concerns the asymptotic gap between VC-dimension and unlabeled compression size for joins; the source does not establish either the existence of the limit or the strict inequality.