This question isn't fundamentally new, but I haven't found one that elicits a clear answer on the conservation of energy problem.
Another question asks: "How Does Hubble's Expansion Affect Two Rope-Tied Galaxies?", and answers, including one from accomplished physicist Luboš Motl, focus on the issue of the scenario being physically invalid.
The rope would eventually break, and maybe it would be slowing the galaxies motion if it were a really tight rope (you can't get rope with the required rigidity to stop the motion of galaxies in Nature) [...] They want to move along the natural trajectories - those we observe - so any rope trying to prevent them from doing so will be stretched by the force of inertia of these galaxies. - Luboš Motl
A rope tethering two galaxies is not only physically complex, but impossible. Furthermore, the answers refer to the rope breaking, and the asker doesn't push them to give a straight answer on whether that's a conservation of energy problem. Motl refers to inertia of the two galaxies being a factor which sidesteps the core issue of the problem, which is the expansion of the two ends of the rope seeming to break the conservation of energy.
I'd like to simplify this question to focus on an extreme, but simpler and more physically valid scenario: A very long rope, by itself.
As space expands, galaxies move away from each-other - atoms in one place move farther away from atoms in another place. So it would stand to reason that the atoms at the two ends of a many trillion lightyear-long rope would move apart, pull away from each-other. In other words, would a force be exerted on the atoms within the rope? Wouldn't that equate to energy created from nothing?
This would seem to break our laws of conservation of energy. If not, why? How would the expansion of space affect a very long rope? Or do we not understand the phenomena well enough to answer?