For a locally closed subscheme $Z\subset X$ (I am interested in the case when $X$ is a variety) one can consider its henselization in $X$ i.e. the 'smallest' pro-etale morphism $U\to X$ such that $Z$ lifts to $U$. We denote the corresponding scheme by $Z_h$.

Now, for $Z_1,Z_2\subset X$ one certainly has a canonical morphism $(Z_1\cap Z_2)_h\to Z_{1h}\times_{(Z_1\cup Z_2)_h} Z_{2h}$. My question is: when can one be sure that this morphism is pro-open? An isomorphism?

It seems that Theorems 2.4 and 2.5 of GRECO S.,STRANO R., Quasi-coherent sheaves over affine hensel schemes (http://www.ams.org/journals/tran/1981-268-02/S0002-9947-1981-0632537-7/S0002-9947-1981-0632537-7.pdf) yield that we have an isomorphism when $X$ and $Z_i$ are affine. I am interested in the case when $Z_i$ are affine, but $X$ is (smooth) projective. So, I have the following questions.

What is currently known about this setting? Actually, I am somewhat confused by Theorems 2.4 and 2.5 mentioned, since their assumptions coincide, whereas their conclusions are very much distinct.

It seems that the argument of Greco and Strano could be carried over to my situation if instead of AIC rings one considers those (irreducible?) pro-varieties whose fraction fields are algebraically closed (see also my previous question Do etale neighhbourhoods of a subvariety descend along base field extensions; does normalization commute with etale base change?). Could this be true, or do I miss something?

Whom could I ask about these matters (in a letter?)? Greco and Strano themselves are not very young at the moment.