Density-one conjecture for weak Carmichael prime factors

For integers $e,f$ with $1\le e\le f$ and $e+f\ge 3$, let ${\mathcal Q}_w(e,f)$ be the set of primes occurring in a weak Carmichael number of the form $p^eq^f$ or $q^ep^f$ for some distinct prime $q$. Density-one conjecture for weak Carmichael prime factors. For every such pair $(e,f)$, ${\mathcal Q}_w(e,f)$ has density $1$ within the set of all primes. This is presented as an open distributional conjecture.