The Choices of the Iliad

In the text we read about how Achilles is consumed by his anger towards Agamemnon and refuses to enter the war. He rejects everyone who tries to compel him into joining the war. Even after many tries Achilles continues to hold on to his anger and resentment. With this being said, do you think that if Patroclus had lived instead of being killed in battle, Achilles would have eventually entered the war anyway? If so, how long do you think that would have taken?