Highly doubt it. The captains still have their captain symbols indicating their status.
I bet the big reveal of episode 9 will be that Ash's classmates are Trial Captains once Ash learns about the Island Challenge. What exactly is preventing Ash from doing the Captain's trials? Nothing. They are all travelling together and they can visit each island for the respective captain's trial.
Well, i edited my post above to add in a bit of reasoning as well.
But i feel that Trial Captain's aren't needed that much actually as part of the story. Since, in-game as well Trial Captain's don't actually have much to do.
In-game you only have battles against Ilima, Lana, Kiawe\Mallow long after their Trial is over. And you only battle Sophocles, who appears as a Challenger to the Champion (aka., the Player). Acerola becomes the new E4 member, and regarding Mina i am completely not sure how hers work.
While the concept of Trial's would still be there, but the part regarding Trial Captain's could easily be filled by some anime only characters or the Island Kahuna.
Aside from that, i think it was mentioned that Kiawe got his Z-Ring after he completed a Trial, so i don't think he is a Trial Captain yet.