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

meaning, does R_a1 equivalence relations have to refine R_a2 equivalence relations? (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 Ra⊑Rl. which means both Ra1⊑Rl and Ra2⊑Rl.

We are currently studying the Myhill-Nerode Theorem so I'm guessing it has to do something with it. I've tried combining few theorems together but came out empty.

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.