Conjecture on algebraic connectivity and Laplacian integrality of power graphs of cyclic groups

Let $n\geq 2$ be an integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $\mathcal{G}(\mathbb{Z}_n)$ denote its power graph. The cyclic power-graph conjecture. The following statements are equivalent:

1. The algebraic connectivity of $\mathcal{G}(\mathbb{Z}_n)$ is an integer.
2. $\mathcal{G}(\mathbb{Z}_n)$ is Laplacian integral.
3. $n$ is a prime power or a product of two primes.

The paper presents this as a conclusion based on the preceding results; the status of the claimed equivalence is not established in the supplied text.