Differential Geometry – Hypothetical uniqueness of embedding a Riemann manifold in a compact kähler

Inspired by this question (isometric embedding of a real analytic Riemann manifold into a compact Kähler manifold), I ask the following:

Accept $ X $ is a true analytic Riemannian manifold with a really real embedding $ X ^ mathbb $ This is Kähler and the Kähler metric is limited to the given Riemann metric $ X $, in addition, $ X ^ mathbb $ is equipped with an antiholomorphic involution whose fixed point set is $ X $, Determine these properties? $ X ^ mathbb $ unique as a germ of manifolds? I think that's true, but I did not find the exact statement in Lempert, Szöke or Guillemin, Stenzel.

Now let it go $ X $ be compact. In response to the quoted question, D. Panov has given some necessary words on how to prove this $ X ^ mathbb $ can be chosen compact. But can $ X ^ mathbb $ Did you choose a canonical style or is it even unique?

In fact, the case interests me even more $ X $ Kähler is manifold with a really real embedding in a hypercounter $ X ^ mathbb $ so that the restriction on the associated Kähler structure is up $ X $ is the given Kähler structure. We also get one $ S ^ 1 $ Action that rotates the complex structures and sets their fixed point set $ X $, Feix and Kaledin have proven that these properties are determinative $ X ^ mathbb $ unique as germ ob $ X $ is completely can $ X ^ mathbb $ be chosen to be complete? Canonically unique? As far as I know, the last questions are far from solved.