I think the fact that in Revelation overcomers are promised to start eating again from the tree of life is evidence that the tree of life is associated both in Eden and in paradise with the maintenance of endless life. The fact that the tree of life is provided by God in the new heaven and the new earth 1000 years after the resurrection, which is when many assume the saints become perpetually immortal, is evidence that access to and eating from the tree of life was necessary for Adam's continuing life even before he sinned.
Why would the promise of eating from the tree of life have any pull power to people who are already permanently immortal?
Rev 2:7 He that hath an ear, let him hear what the Spirit saith unto the churches; To him that overcometh will I give to eat of the tree of life, which is in the midst of the paradise of God.
Rev 21:1 And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.
Rev 22:1 And he shewed me a pure river of water of life, clear as crystal, proceeding out of the throne of God and of the Lamb.
Rev 22:2 In the midst of the street of it, and on either side of the river,
was there the tree of life, which bare twelve
manner of fruits,
and yielded her fruit every month: and the leaves of the tree
were for the healing of the nations.