Prime-index density-zero conjecture for reducible cyclotomic compositions

Let $\bar D(t)$ count good pairs $(m,n)$ with $m,n$ both prime and $m\leq t$, $n\leq t$, where goodness means that $\Phi_m(\Phi_n(x))$ factors over the integers. Let $\pi(t)$ denote the number of primes at most $t$. Prime-index cyclotomic composition density conjecture.

$$\lim_{t\to\infty}\frac{\bar D(t)}{\pi(t)^2}=0.$$

This is the prime-index analogue of the preceding density conjectures, restricting attention to cyclotomic polynomials indexed by primes.