  In this paper we consider the limiting distribution of KPZ growth models with
random but not stationary initial conditions introduced in
[Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given
in terms of a variational problem. By directly studying it, we deduce the right
tail asymptotic of the distribution function. This gives a rigorous proof and
extends the results obtained in [Meerson-Schmidt 2017].
