In katz's paper "Serre-Tate local moduli" chaper 3 has the following construction:
Let $A$ be a fixed ordinary elliptic curve defined over $k$ of char $p>0$. Consider the deformation of $A$ to $W(k)$-witt vector ring. Denote by $\mathcal{M}_A$ the corresponding formal moduli space,  and let $\mathcal{A}/\mathcal{M}_A$ denote the universal formal deformation of $A/k$. 
Let $R$ be any artin local ring with risidue field $k$, and any lifting $\mathbb{A}/R$ of $A/k$. Let $\mathbb{A}'/R$  be the quotient of $\mathbb{A}$ by the "canonical  subgroup"  $\hat{\mathbb{A}}[p]$ of $\mathbb{A}$. Here $\hat{\mathbb{A}}$ denote the connected part of the $p$-divisible group associated to $\mathbb{A}$.


If we apply the construction 
$$
\mathbb{A}/R\to \mathbb{A}'/R
$$
to the universal formal deformation $\mathcal{A}/\mathcal{M}_A$, we obtain a formal deformation  $\mathcal{A}'/\mathcal{M}_{A^{(\sigma)}}$ of $A^{(\sigma)}/k$, here $\sigma$ is Frobenius.

I have some question about this $\mathcal{A}'/\mathcal{M}_{A^{(\sigma)}}$.
I guess that it has the following property:
Let $W_n(k)$ denote the witt vector ring of length n. And let $j_n$ denote the natural closed iimmersion  $spec(W_{n-1}(k))\to spec(W_n(k))$.
suppose we have two  $W_n(k)$ point of $\mathcal{M}_{A^{(\sigma)}}$: for $i=1,2$

$ s_i:spec(W_n(k))\to\mathcal{M}_{A^{(\sigma)}}$.

so we obtain two  Elliptic curves $E_i$ over $W_n(k)$ .

Then we have the follwoing statement:

If $s_1\circ j_n=s_2\circ j_n$. Then $E_1$  and $E_2$ should be isomorphic over $W_n(k)$.

is this statement  true? 
Thank you!