ag.algebraic geometry - Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth commutative group scheme over $k$. Suppose now that $B_k$ is an abelian variety over $k$ that is a quotient of $A_k$.

Does there exist an abelian scheme $B$ over $R$ and a morphism
$Arightarrow B$ of smooth commutative $R$-groups whose base change to $k$ is the quotient mapping $A_k rightarrow B_k$?

The answer is NO, I'm fairly sure, as the generic fiber of $A$ could be simple with
$A_k$ an extension of an abelian variety by a torus. Let us
therefore assume that $B_k$ can be lifted up to isogeny, i.e. that there exists $C$
an abelian scheme over $R$ such that 1) $C_k$ is isogenous to $B_k$ and
2) There exists a map of smooth groups $Arightarrow C$ over $R$ whose base change to $k$
is the quotient map $A_krightarrow B_k$ followed by the isogeny $B_krightarrow C_k$. With this added assumption, is the answer to the question above still NO?

I'm inclined to think that this is the case, but can't immediately convince myself of this.

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Reformulation

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:

1. $B_K$ has good reduction


2. Any abelian variety quotient $A_Krightarrow C_K$ of $A_K$ having good reduction
   factors through $A_Krightarrow B_K$.

If we impose the additional assumption that the kernel of $A_Krightarrow B_K$ 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 $A_Krightarrow B_K$ induces a mapping $Arightarrow B$ 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 $av(A^0_k)$?