Maybe there is a simpler way to state the idea:.
1; would the two (planetary) cultures diverge?
(yes, for a variety of easy reasons).
2; would this divergence become more significant over time?
(yes, as at least some of any differences will inherently be amplified by multiple factors on multiple levels of multiple types of process for multiple reasons over multiple hundreds to thousands of years, and that differences in any one planetary cultural/functional aspect tend to create and become entangled with differences in multiple other cultural/functional aspects).
3; would the degree of divergence, over time, eventually become significant—ie, in the sense that it results in some sort of 1st strike game-theory dynamic?
(yes, insofar as cultural development differences cannot not also be fully entangled with technological developmental differences).
So then the question becomes: “is it even possible to maybe somehow constrain any or all of these three process factors to at least the minimum degree necessary so as to adequately prevent that factor, and thus of the overall sequence, from occurring?”.
In regards to this last question, after a lot of varied simplifications, it becomes eventually and finally equivalent to asking: “can any type of inter-planetary linear causative process (which is itself constrained by speed-of-light latency limits) ever fully constrain (to at least the minimum degree necessary) all types of non-linear local (ie; intra-planetary) causative process?”.
And the answer to this last question is simply “no”, for basic principled reasons.
The claim is there would be some differences, in the short term (dozens of years), and that these would necessarily _become_ huge discrepancies, over the long term (hundreds or thousands of years). This amplification of difference is inevitable due to multiple nonlinear effects operating in all types of life ecosystems, cultures, etc. A few thousand years is a trivial timescale when it comes to planetary evolution dynamics and life cycles, which is generally measured in billions of years.