Analyzing the rocket-equation section, I found the following statement:

The relativistic rocket equation is

$Δ v = c tanh (\frac{I_{s p}}{c} ln \frac{m_{0}}{m_{1}})$

where $Δ v$ is the difference in velocity, $m_{0}$ is the initial mass of the probe, $m_{1}$ the final mass of the replicator and the $I_{s p} / c$ term denotes the specific impulse of the fuel burning process. The $I_{s p} / c$ term can be derived from η, the proportion of fuel transformed into energy during the burning process $I_{s p} / c = \sqrt{2 η - η 2}$ [24].

The fact that we can fully derive $I_{s p}$ from the fuel energy-transformation efficiency seemed weird to me, so I looked it up in the underlying reference and found the following quote (I slightly cleaned up the math typesetting and replaced it with equivalent symbols above, emphasis mine):

For the relativistic case, there is a maximum exhaust velocity for the reaction mass that is given by:

$w = \sqrt{η (2 - η)} c$

where e is the fuel mass fraction converted into kinetic energy of the reaction mass. I was not able to improve on that derivation from a presentation point of view.

This is obviously a very similar equation, but importantly this equation specifies an upper bound on the exhaust velocity and does not say when that upper bound can be attained. Intuitively it seems to me not at all obvious that we should be able to attain that maximum exhaust velocity, since it would require the ability to perfectly direct the energy released in the fuel burning, which would naively be primarily released as heat.

The keyword that finally helped me understand the relevant rocket design is "Fission-Fragment Rocket", which at least according to Wikipedia could indeed reach specific impulses sufficient to support the conclusions in the paper.