You say you're willing to change your mind based on where the evidence leads. I challenge you to follow the evidence about immortality. What evidence is there *besides the fact we can imagine it and it feels very real*?
In response to evidence of immortality, I'd like to propose this perspective. First, I'd like to change the word "immortality" to "afterlife" for the purpose of this perspective. I think immortality assumes one or similar forms while afterlife signifies a beginning and an end. I'd like to extend that thought to reincarnation to further my perspective.
Let me start this perspective by providing my "evidence".
It's obvious that we're all here, functioning, conscious, sentient beings living on a physical plane. This is my evidence.
What happens when we die? Nobody knows. But, would it be too far of a leap to say that we are re-born into some life form whether here, there, anywhere in reality? The proof is that we're all here. We may have come from previous lifetimes and may be going to future lifetimes, albeit, maybe in a different vessel in a totally different part of the Universe.
When you think about it, we know nothing else. So, who's to say that when we die, we aren't in a sleep-like state. Time as we understand it simply goes out of our perception, so it may be 1 millisecond before we're transported into a new vessel and start life as a baby somewhere else in this grand Universe, or it maybe Trillions of expansions and contractions of our Universe before we're re-born. We don't know and we won't be able to "feel" the expanse of time because we won't have any perception of it, but we'll be born again into another physical vessel of life.
For instance, when you fall into a deep sleep and if you don't have very rigid patterns of sleep and wake up times, you don't know whether you've slept 4 hours or 8 hours or 10 hours when you awake. It just feels like you went to sleep and then woke up. Your perception of time is dulled. The comedian Martin Lawrence related that he once went jogging in one of those heavy sweat suits and fell into a coma. He slept for 3 days and woke up not knowing that much time had passed. Everyone around him was super worried and thought he was going to die. When he woke up 3 days later, he just thought he had a really good night's sleep.
So, what do you think? If we have no perception of time once we leave our bodies, and our existence is merely a random chance, and time is eternal, and according to statistics, this random chance will indeed happen again throughout infinity, and we obviously have consciousness and are living . . . wouldn't it be logical to say that we'll be reincarnated at some point in time?