In the New Testament, the story of Jesus' activities is told four times, in the gospels of Matthew, Mark, Luke, and John. In John, and not the others, there is a person referred to repeatedly as "the disciple whom Jesus loved." No name, just "the disciple whom Jesus loved."

Conservative Christians are sure what that doesn't mean.

I am not sure. It might mean Jesus had a clear best friend. It might mean that after Jesus died, someone wanted to sell the idea that he was Jesus' best friend, even though he wasn't. It might mean that Jesus who walked among jostling crowds and healed people, body and soul, got to sometimes spend time with his friend and feel great, body and soul.