I think that will happen. Like say Obito is close to losing, he probably yelled his chance to see Rin won't end this way, and Naruto may realize that this Rin person was something "special". Naruto read his chakra and according to him, he's suffering. He needs to know why he's going so far if he can just stood by with his friends. Right now, nothing phasing him because he only got Hokage part down, but he needs the Rin part down.
But my question is , if Obito tells about Rin and things, what will be Naruto's answer ?