Carmichael's fixed-prime-factor infinitude conjecture

A Carmichael number is a composite number $n$ satisfying

$$a^n\equiv a\pmod n$$

for every $a\in\mathbb Z$. For $R\in\mathbb N$ with $R\geq 3$, consider Carmichael numbers having exactly $R$ prime factors.

Carmichael's fixed-prime-factor infinitude conjecture. For any $R\in\mathbb N$ with $R\geq 3$, there exist infinitely many Carmichael numbers with $R$ prime factors.

Carmichael numbers are known to be infinite, but the existence of infinitely many with each prescribed number of prime factors remains a central open problem. The paper obtains conditional results toward related questions.