Monotonicity conjecture for cyclic 321-avoidance growth

For each $n\geq 1$, let $c_n=|\mathcal C_n(321)|$, where $\mathcal C_n(321)$ is the set of cyclic permutations of size $n$ avoiding $321$. The monotonicity conjecture. The sequence

$$\sqrt[n]{c_n}$$

is monotonically increasing in $n$. The paper has already established that its limit exists by supermultiplicativity, while the asserted monotonicity is presented as a stronger unproved result.