Faltings height in short exact sequences

I think the following should give a counterexample. Let $\mathcal{O}$ be an order in an imaginary quadratic field $K$ and $\mathcal{O}_K$, the ring of integers. Then it's not too hard to find a (non-split) short exact sequence of $\mathcal{O}$-modules: $$0 \to \mathcal{O}_K \to \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}_K \to 0,$$

e.g. if $1, \omega$ is a basis of $\mathcal{O}_K$, with $\omega^2 \in \mathbb{Z}$, then send $(a,b)$ to $\omega a - b$. If $A$, $B$, and $C$ are the abelian varieties (over $\mathbb{C}$) corresponding to these lattices (so $A = \mathbb{C}/\mathcal{O}_K$, etc.), then $$0 \to A \to B \to C \to 0.$$

Indeed, the maps on lattices induce $\mathbb{C}$-linear maps on complex vector spaces which preserve the lattices, so you get maps $A \to B \to C$. $B \to C$ is clearly surjective, and $A \to B$ is injective because $\mathcal{O}_K$ (the cokernel of the map of lattices) is torsion-free. Exactness in the middle you can check by hand.

If the Faltings height is additive then this exact sequence of abelian varieties gives that $h(\mathbb{C}/\mathcal{O}) = h(\mathbb{C}/\mathcal{O}_K)$, where I really mean to take the heights of models over a number field. But in general $h(\mathbb{C}/\mathcal{O}) \neq h(\mathbb{C}/\mathcal{O}_K)$, as can be seen from the formulas on pages 273-274 of this paper by Tonghai Yang.

Proposition 3.3 of Ullmo's paper "Hauteur de Faltings de quotients de J_0(N) " (American Journal of Math., 2000) seems to answer your question.