Classification of all integers dividing their shifted sigma values

Determine whether the set of positive integers b such that b divides σ(b^k − 1) for every positive integer k is exactly {1, 3, 4, 6, 8, 12, 24}, with no other values of b satisfying this property.

Background

The paper establishes divisibility results for σ(zk − 1) for the values z = 3, 4, 6, 8, 12, and 24, and then identifies the additional value 1 in the proposed complete set B = {1, 3, 4, 6, 8, 12, 24}. It explicitly leaves unresolved whether these are the only values of b for which b divides σ(bk − 1) for every k ∈ N. The conjecture asks for a complete classification of all such integers.