$\newcommand{\F}{\mathbb{F}} 
\newcommand{\End}{\mathrm{End}} 
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}$
I would like to know if the following is true:

> **Proposition A** : Let $A_1, A_2$ be two abelian varieties over a finite field $k$. If $\End_{\overline k}(A_1) \otimes_{\Z} \Q$ and $\End_{\overline k}(A_2) \otimes_{\Z} \Q$ are isomorphic (as $\Q$-algebras), then $A_1, A_2$ are isogenous over $\overline k$. 




The converse holds: let $\phi : A_1 \to A_2$ be an isogeny of degree $m$. There is an isogeny $\psi : A_2 \to A_1$ such that $psi \circ \phi = [m]$ (see Poonen "Lectures on rational points", Proposition 4.1.19 ; this is not the dual isogeny!). Define a map $\End_{\overline k}(A_1) \otimes_{\Z} \Q	\to	\End_{\overline k}(A_2) \otimes_{\Z} \Q$ via
 $f   \longmapsto 	  \dfrac{1}{m}  \phi \circ f \circ \psi	$.
 It is an algebra isomorphism. 


The result holds over $\mathbb C$, see [here][1] or [Prop. 1.2.17][2]. It holds for elliptic curves over a finite field: the supersingular case can be proved "by hand", using Tate isogeny theorem, while the ordinary case follows from Deuring's work on CM (lifting, etc.).
Note that the curves do *not* need to be isogenous over the base field $k$. I skimmed through the paper _Endomorphisms  of  Abelian  Varieties  over  Finite  Fields_ of Tate, but I did not find such a statement.

Maybe one could use Serre--Tate theory instead of Deuring, but it seems to be only available for ordinary abelian varieties. I am not aware of the details here anyway. If the claim does not hold, is there a suitable hypothesis to make it true?


  [1]: https://mathoverflow.net/questions/158410/abelian-varieties-with-the-same-cm-type-are-isogenous
  [2]: http://iml.univ-mrs.fr/~kohel/phd/thesis_wilson.pdf