Conjecture on the logarithmic density of non-prime endomorphisms of the free group

Let $F_2$ be the free group of rank two, let $B_n$ be its ball of radius $n$, and let

$$\mathcal N_n=\{(u,v)\in B_n\times B_n:\varphi_{u,v}\text{ is not prime}\}.$$

Define the logarithmic density of non-prime pairs by

$$\delta_{\mathrm{np}}=\limsup_{n\to\infty}\frac{\log |\mathcal N_n|}{\log |B_n\times B_n|}.$$

Conjecture on the non-prime density. The logarithmic density satisfies

$$\delta_{\mathrm{np}}=\frac34.$$

The main results establish the bounds $\frac34\leq\delta_{\mathrm{np}}\leq\frac{27}{28}$, so the conjecture asserts that the lower bound is sharp. Determining the exact density measures how large the exceptional family of non-prime endomorphisms is among pairs of elements in the free group.