Generalized prime-index density-zero conjecture for reducible cyclotomic compositions

Let $f(t)$ and $g(t)$ be strictly increasing functions tending to infinity as $t$ tends to infinity. Let $\bar D_{f,g}(t)$ count good pairs $(m,n)$ with $m,n$ prime, $m\leq f(t)$, and $n\leq g(t)$, where goodness means that $\Phi_m(\Phi_n(x))$ factors over the integers. Generalized prime-index cyclotomic composition density conjecture. For any such $f(t)$ and $g(t)$,

$$\lim_{t\to\infty}\frac{\bar D_{f,g}(t)}{\pi(f(t))\pi(g(t))}=0.$$

This is the function-dependent prime-index analogue of the preceding density conjectures and is stated as an expected extension.