Potential FTL Implications of quantum “weak measurement”?

Physics noob here. I can't find a good answer to this thought experiment that is consistent with general relativity and It's really bothering me. I'm probably missing something huge but just can't figure it out, so here it goes.

If many particle pairs are entangled on Earth and one particle of each of the pairs of particles is transported to a distant point in space at sub-light speeds using any means, then wouldn't faster than light information transmission be possible between those pairs? The particles on Earth could be measured along a particular spin axis, forcing the other particles to take the opposite spin along that axis. Using "weak measurement" couldn't a measurer at the distant point in space statistically determine the wave function state and therefore when to check for spin axis alignment and therefore a transmitted binary value? I have a feeling the answer may be "weak measurement isn't possible" but I don't know enough to say whether the literature to date has reached a consensus.

In quantum mechanics, systems can only affect one another through local interactions. In an entangled pair of systems $S_1,S_2$, $S_1$ contains information about $S_2$ but that information can't be revealed by any measurement on $S_1$ alone. The information is only revealed when then systems containing measurement results from $S_1$ interact with those containing measurement results from $S_2$ and that is when the correlations between measurement results are established:

https://arxiv.org/abs/quant-ph/9906007

So the correlations are established through interactions, which are local since the equations of motion in quantum mechanics are local.

A weak measurement is just a measurement that involves a weak coupling between the measured system and the measurement device. Such a measurement obeys the same laws as any other kind of measurement and so can't result in non-local influences between systems.