  We continue our study of the scale-inhomogeneous Gaussian free field
introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free
energy on V_N and adapt a technique of Bovier and Kurkova (2004b) to determine
the limiting two-overlap distribution. The adaptation was already successfully
applied in the simpler case of Arguin and Zindy (2015), where the limiting free
energy was computed for the field with two levels (in the center of V_N) and
the limiting two-overlap distribution was determined in the homogeneous case.
Our results agree with the analogous quantities for the Generalized Random
Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova
(2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra
identities hold exactly in the limit. As a corollary, the limiting array of
overlaps is ultrametric and the limiting Gibbs measure has the same law as a
Ruelle probability cascade.
