Products of test elements of free factors of a free group

Let G be a group. An element g∈G is called a test element of G if for every endomorphism φ:G→G, φ(g)=g implies that φ is an automorphism. Let F(X) be a free group on a finite non-empty set X, and let X=X1∐X2∐…∐Xr be a finite partition of X into r≥2 non-empty subsets. For i=1,2,…,r, let ui∈〈Xi〉≤F(X), and let w(z1,…,zr) be a word in the variables z1,…,zr. We give several sufficient conditions on ui (1≤i≤r) and w for w(u1,…,ur) to be a test element of F(X). As an application of these results, we give examples of test elements of a free group of rank greater than two that are not test elements in any pro-p completion of the group.