dg.differential geometry - Extension of strictly plurisubharmonic functions on a Kähler manifold

It is instructive to conisder the case of Kahler metrics invariant under torus action. In this case your question becomes a certain (nontivial) question on convex functions.

Recall first, that Kahler metrics on $(mathbb C^*)^n$ invariant under the action of $(S^1)^n$ have global potential that is given by a convex function $F$ on $mathbb R^n$. Here $mathbb R^n$ is identified with the quotient

$(mathbb C^*)^n/(S^1)^n$ and we take coordinates $log|z_i|$ on $mathbb R^n$.
So we can translate your original question as follows

QESTION. Suppose you have a smooth convex function $F$, defined on $mathbb R^n$ outside compact $Omega$. Is it possible to extend $F$ to a smooth convex function on the whole $mathbb R^n$?

It easy to construct an example of a non-convex $Omega$ on $mathbb R^2$, with convex $F$ defined on $mathbb R^2setminus Omega$, so that $F$ can not be extended. For the moment I don't see how to make such an example when $Omega$ is the unite disk, but it sounds plausible that such examples exist.