Given two DFA's accepting the same language, does one have to refine the other? I have a logical question that I can't quite crack:

Given two automata accepting the same language $L$, does one have to refine the other?

In other words, if $A_1$ and $A_2$ both accept $L$, with associated equivalence relations $R_{A_1}$ and $R_{A_2}$, does $R_{A_1}$ have to refine $R_{A_2}$, or vice versa?
I am leaning toward the answer yes because if we have a regular language $L$ which is accepted by an automaton $A$, we can show that the relation $R_A$ refines the relation $R_L$, meaning $R_A \sqsubseteq R_L$,
which means that both $R_{A_1} \sqsubseteq R_L$ and $R_{A_2} \sqsubseteq R_L$.
We are currently studying the Myhill-Nerode Theorem, so I'm guessing it has something to do with it.
I've tried combining few theorems together, but came out empty.
For an integer $n$, consider the following DFA $A_n$ on the alphabet $\{a\}$. The set of states is $\{q_0,\ldots,q_{n-1}\}$. The initial state is $q_0$. All states are accepting. The transition function is $\delta(q_i,a) = q_{i+1}$, where we identify $q_n$ with $q_0$.
Do the relations of $A_2$ and $A_3$ refine each other?