Generalized Erdős–Sierpiński conjecture for the sum-of-divisors function

Let $k$ be a positive integer, let $n$ range over positive integers, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Generalized Erdős–Sierpiński conjecture. For any positive integer $k$, the equation

$$\sigma(n + 1) = k\sigma(n)$$

has infinitely many solutions. The case $k=2$ is proved in the paper assuming Schinzel's H Hypothesis, while the assertion for arbitrary positive integers $k$ is posed as the generalization and remains open.