Bajnok's limit conjecture for cyclic-group independence numbers

Let $t\geq 2$ and let $\tau(\mathbb{Z}_n,t)$ denote the corresponding independence parameter for the cyclic group $\mathbb{Z}_n$. Consider the normalized quantity

$$\frac{\tau(\mathbb{Z}_n,t)}{n^{1/\lfloor t/2\rfloor}}.$$

Bajnok's limit conjecture. The value of

$$\lim \frac{\tau(\mathbb{Z}_n,t)}{n^{1/\lfloor t/2\rfloor}}$$

exists if and only if $t$ is even. This concerns the asymptotic behavior of the independence parameter between the bounds established immediately beforehand; the conjecture leaves the odd values of $t$ unresolved.