Conflicts and participants in the Player's Turn

When one of the players decides to start a conflict during the players’ turn by paying a mark or with his free test, can the other players participate in that conflict for free to help the other player, or do they have to spend a check to get involved?
I know that if another character starts a conflict against you you don’t have to pay, but in case you want to participate in a conflict on the side of the mouse that spends the mark to start the conflict, do you have to pay as well?
If you can participate for free, do you also bind yourself to the twists and conditions that may arise in the compromises?

I hope I have explained myself well.
Lately I am finding too many holes in the rulebook during my games.

Fellow patrol mates can join the scene freely. They do become entangled in the results. So, if it were just a simple test or Vs test, fellow patrol mates can offer Helper dice or Wises. For a Conflict, fellow patrol mates can join without spending a Check, can offer Helper dice or Wises, and can roll for the Conflict actions.

In the case of a simple test or Vs test, those Helpers are bound to the results while the Wises are only impacted by a Twist. In the case of a Conflict, all participants are bound to the results.

Thanks!!!

So the Mouse who is starting the conflict and paying the check must be the captain on his team I guess, and the one is rolling disposition

Yes; if the player used the check, they are leading the scene.