Infinitude of prime and lucky numbers of every fractional order

Under the restricted factorization described in the source, assign a fractional order $\operatorname{ord}(p)$ or $\operatorname{ord}(l)$ recursively to primes $p$ and lucky numbers $l$: lucky primes have order $1$, products of powers of lucky primes have prime order $1/2$, and products of primes have lucky order $1/2$; in general, the order is determined by the smallest fractional order of the relevant factors as specified by the recursive definition. Fractional-order infinitude conjecture. For every fixed positive integer $n$, there are infinitely many prime numbers or lucky numbers of order $1/n$. This generalizes the conjecture that there are infinitely many lucky primes, but the source gives no resolution.