Asymptotic density of separable numbers

Determine whether A(x) = o(x) as x → ∞, where A(x) denotes the number of positive integers n ≤ x that are separable, meaning there exists a positive integer m such that m and n form an interlocking pair (between every two divisors greater than 1 of each number lies a divisor of the other).

Background

Erdős and Hall introduced interlocking pairs (m, n) and defined a positive integer n to be separable if it belongs to such a pair. They denoted by A(x) the count of separable integers up to x and showed a lower bound A(x) > c x / log log x for large x, but were unable to determine the exact asymptotic order.

This paper resolves two other conjectures by Erdős and Hall (about powers of two and products of the first primes) but explicitly notes that the question of whether A(x) = o(x) remains open and is considered the most important open problem concerning separable numbers.