If I understand correctly, Alice and Bob generate an entangled pair of photons and each take one. Alice does something with hers (you specified a quantum eraser experiment at a particular time, but I won't make that assumption) while Bob does a standard double-slit experiment with his.
A minor point here is that with one photon you'll never get interference fringes, just a dot, but they can work around that by repeating the experiment many times, with Alice trying to send the same information each time, until a pattern has built up on Bob's screen.
The outcome of this experiment depends on whether any measurement of Alice's photon could reveal which slit Bob's photon went through. (A more precise description of the experimental setup would settle this question.) If it can, then Bob will not see an interference pattern no matter what Alice does. This is because for the interference pattern to disappear it's only necessary that which-path information be recorded somewhere and in principle accessible to measurement; it doesn't matter whether the measurement actually happens.
If it can't, then Bob will see an interference pattern no matter what Alice does, because any measurement Alice could perform will not collapse the part of the wave function related to the two slits. Collapse isn't all or nothing; only the part of the wave function associated with the measured quantity collapses.
These two cases are actually two ends of a continuum; if measurements on Alice's photon can reveal partial information about the path, then Bob will see a pattern intermediate between the fully interfering and non-interfering patterns. But in no case does the pattern depend on what Alice actually does with her photon, only on what it could tell her in principle.
In the paper "Time-resolved double-slit experiment with entangled photons" (mentioned in your answer), although the text for Fig. 4 says "Interference pattern fringes move as the phase is changed remotely by the QWP", it appears that they are talking about fringes that only appear after postselecting with D1 or D2, as in the usual quantum-eraser setup. Note that the fringes in Fig. 4(B/D) are labeled "heralded by D1/D2", and Fig. 4(A) shows a total detection envelope with no fringes.
I only read the abstract of the paper mentioned in your question, but it only alleges a violation of the principle of complementarity. Unlike the uncertainty principle, the complementarity principle isn't a foundational principle of quantum mechanics and never had a standard mathematical formulation. If a particular attempt to formalize it turns out to be wrong, that's not a problem for quantum mechanics.