<PERSON>'s models do not work because we live in a phase with a positive cosmological constant. This leads to a de Sitter horizon which limits the possible growth of computational capacity, and the best we can hope for is Poincare recurrence assuming causal horizon complementarity. Besides, if superstring theory were true, our nonsupersymmetric phase can only be at best metastable. Its lifetime might be exponentially long, but that's still finite.
<PERSON> and <PERSON> are working on the Census Taker's Hat, which might be just what you are asking for. In the landscape of string theory, there exists supersymmetric vacua with exactly zero cosmological constant. It's possible for our phase to tunnel to such a vacuum in the future. The future conformal boundary of the new phase will be a null "hat". A Census Taker with eternal life will enter the new phase and grow without limit and finally end up at the Census Bureau at future conformal infinity. During its lifetime in the new phase, the Census Taker will emit massless radiation heading off to the Census Taker's Hat (i.e. 419
future null infinity). This will continually transmit information from the Census Taker to the Hat where they will be recorded holographically. In this article by <PERSON> and <PERSON>, they claim that
Over time, the Census Taker receives an unbounded amount of information, larger than the entropy bound on any of the finite causal diamonds beyond the hat. This means that the Census Taker will receive information about each patch history over and over again, redundantly.
and this will happen infinitely many times.
Unfortunately, information represented holographically at the Hat will be frozen, and to give them eternal life, there needs to be someone outside our universe to read and interpret them. This can happen if our universe is really a computer program running a limit computable simulation. After some finite time, this simulation will be stopped and its output processed. This corresponds to an ordinal jump. To have eternal life, we need infinitely many ordinal jumps all the way to the ordinal of all ordinals, which is itself not an ordinary ordinal. We have to push past the uncountable $\aleph_1$, but this is fine because of the Lowenheim-Skolem theorem ensuring the existence of nonstandard countable models. A limited weakened form of the axiom of countable choice is guaranteed by the probabilistic interpretation of quantum mechanics. 419