Generalized infinitude conjecture for weak Carmichael numbers

Let $s\ge 2$ be an integer, and let $(e_1,e_2,\ldots,e_s)$ be a fixed tuple with $e_1\ge e_2\ge\cdots\ge e_s\ge 1$ and $\sum_{i=1}^s e_i\ge 3$. Generalized weak Carmichael infinitude conjecture. There are infinitely many weak Carmichael numbers

$$n=p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s},$$

where $p_1,p_2,\ldots,p_s$ are distinct odd primes. This generalizes the conjectured infinitude of the two-prime families.