# Height variation of abelian varieties within an isogeny class

Let $$A$$ be an abelian variety defined over a number field $$K$$ of dimension $$g \geq 2$$, and put $$h_F(A)$$ for the (stable) Faltings height of $$A$$. It is well-known from the seminal paper of Faltings that for a given $$A$$ there are only finitely many abelian varieties $$B$$ which are isogenous to $$A$$ over $$K$$. Note that the Faltings height is not an isogeny invariant but is bounded within the isogeny class. Further, $$h_F(A)$$ is in fact a $$\overline{K}$$-isomorphism invariant.

Let $$\mathfrak{C}(K,g)$$ be the set of isogeny classes of abelian varieties $$A$$ defined over $$K$$ of dimension $$g$$. For a given class $$C \in \mathfrak{C}(K,g)$$, let $$S(C)$$ be a set of representatives of $$K$$-isomorphism classes in the class $$C$$. By Faltings' theorem, $$S(C)$$ is finite. Thus the following is well-defined and finite:

$$\displaystyle N(C) = \sup_{A,B \in S(C)} |h_F(A) - h_F(B)|.$$

Is the quantity $$N(C)$$ bounded as $$C$$ varies over $$\mathfrak{C}(K,g)$$?

A related question: is it expected that $$\# S(C)$$ is bounded as $$C$$ varies over $$\mathfrak{C}(K,g)$$?