Generalized perfect-number conjecture for Mersenne exponents

Let $k>2$ be a prime such that $2^k-1$ is a Mersenne prime. For integers $\alpha, \beta>1$ and an odd prime $p<3\cdot 2^{\alpha-1}-1$, let

$$n=2^{\alpha-1}p^{\beta-1}.$$

Generalized perfect-number conjecture. One has

$$n\mid \sigma_k(n)$$

if and only if $n$ is an even perfect number different from $2^{k-1}(2^k-1)$.

This conjecture extends the preceding results for $k=3$ and $k=5$ to primes $k$ for which $2^k-1$ is a Mersenne prime. The source states that computation supports it, while the described method does not apply to other values of $k$; its general status is therefore open.