  Let $(\pi,\sigma)$ traverse a sequence of pairs of cuspidal automorphic
representations of a unitary Gan--Gross--Prasad pair $({\rm U}_{n+1},{\rm
U}_n)$ over a number field, with ${\rm U}_n$ anisotropic. We assume that at
some distinguished archimedean place, the pair stays away from the conductor
dropping locus, while at every other place, the pair has bounded ramification
and satisfies certain local conditions (in particular, temperedness). We prove
that the subconvex bound \[ L(\pi \times \sigma,1/2) \ll C(\pi \times
\sigma)^{1/4 - \delta} \] holds for any fixed \[ \delta < \frac{1}{8 n^5 + 28
n^4 + 42 n^3 + 36 n^2 + 14 n}. \] Among other ingredients, the proof employs a
refinement of the microlocal calculus for Lie group representations developed
with A. Venkatesh and an observation of S. Marshall concerning the geometric
side of the relative trace formula.
