Auil's conjecture that the iterative sequence enumerates the Möbius numbers

Let $\mathbb K=\{k_i\}_{i\in\mathbb N}$ be the set of integers produced by the paper's iterative construction, and let

$$\mathbb M=\{m\in\mathbb N:\mu(m)\neq0\}$$

be the set of square-free, or Möbius, numbers. Auil's enumeration conjecture. The iterative construction produces every Möbius number and no others:

$$\mathbb K=\mathbb M.$$

The initial terms displayed in the source agree with the square-free numbers, and numerical evidence is reported in support of the conjecture. Its general validity remains unresolved.