Just for the sake of more silliness in this thought experiment. I believe with your tube (and with a spacesuit) that wasn't in a vacuum you would reach terminal velocity but by the time you reached the center you would be weightless and never be able to go out the other side since the gravity component, a key component of terminal velocity (as is drag), is practically a net zero.

That tube would need a really long ladder to get to the other side. In my example, only the friction from air is removed (drag) leaving the gravity vector, but since you never reach terminal velocity, you would shoot out the other end. With the gravity vector still in play, you would pop out at the height on the other side the same as the height where you jumped in.

I'm not sure how long it would take in your example, but I think around 32 hours just to get to the center (3,959) if going at terminal velocity. That's just spitballing because I'm not that strong in mathematics. Thanks for making me think about this too much, Dan!