Infinite weak Carmichael families with a fixed prime

For integers $e,f$ with $1\le e\le f$ and $e+f\ge 3$, and an odd prime $q$, let ${\mathcal Q}_w(q;e,f)$ be the set of primes $p$ such that $p^eq^f$ or $q^ep^f$ is a weak Carmichael number. Infinite weak Carmichael family conjecture.

$$\bigcup_{1\le e\le f<\infty}{\mathcal Q}_w(q;e,f)$$

is infinite. The claim predicts infinitely many compatible weak Carmichael constructions for each fixed odd prime $q$.