Bologna were a man and goal behind but fought back to win 2-1 on Sunday.
Conte snapped: “The only thing I can say is that from here to the end of the season each of us, including me, have to prove we belong at Inter. That's true for the coach and all the players.
“If there is a winning project and we want to get back to winning, we must understand if this is the right coach to win and the right players to win.
“It's going to be an examination for everyone to the end of the season. We've got to give our all, knowing we are up for discussion. I am first in line, as is only right."
On their title challenge, he added: “After gifting a game like this, and also the draw with Sassuolo, it becomes difficult. We have to secure the Champions League place as soon as possible and try to make fewer mistakes.
“I repeat, we all have to prove that we can be part of a winning project, everyone from the coach to the last person on the Inter salary."