Consider the following theorem of Chevalley (see 9.2/1 of the book "Neron Models" by Bosch, Lutkebohmert and Raynaud):
Theorem: Let $k$ be a perfect field and $G$ a smooth and connected algebraic $k$-group. Then there exists a smallest connected linear subgroup $L$ of $G$ such that the quotient $G/L$ is an abelian variety. Furthermore, $L$ is smooth and of formation compatible with extension of $k$.
Definition: We write $av(G)$ for $G/L$ as in the Theorem.
Now fix a dvr $R$ of mixed characteristic $(0,p)$ with fraction field $K$ and residue field $k$. Let $A_K$ be an abelian variety over $K$. There exists an abelian variety quotient $B_K$ of $A_K$, unique up to isogeny, with the following properties:
If we impose the additional assumption that the kernel of $A_K\rightarrow B_K$ is connected (i.e. an abelian sub-variety of $A_K$), then $B_K$ is uniquely determined. We call this $B_K$ the maximal good reduction quotient of $A_K$.
The surjection $A_K\rightarrow B_K$ induces a mapping $A\rightarrow B$ on Neron models over $R$ and hence a mapping on identity components of closed fibers $A^0_k \rightarrow B_k$ which yields a homomorphism of abelian varieties $$\varphi:av(A^0_k)\rightarrow B_k.$$
Question: Is the kernel of $\varphi$ an abelian sub-variety of $av(A^0_k)$?