I don't think acid free boards do anything like trap and neutralize acids. They are just a barrier say between other posters, and being acid free, the board shouldn't contribute to adding more acid into the poster
Another related thought, PVC tubes for containing rolled posters is supposed to be not good as gases leak out affecting posters somehow
The acid not only browns the poster, it makes it more brittle over time so the poster over time will turn to confetti, starting with tearing. Linen backing with washing and de-acidifying, (backing also helps protect against dings/bump damage and damage from repeatedly folding and unfolding ) plus keeping it away from direct and reflected sunlight (UV) and keep in a constant temperature environment I think would be the way to preserve it best, even though linen backing does its own damage, supposedly is irreversible due I assume to it getting stretched a bit in the process and saturated with glue paste.
If I see a poster for sale showing evidence of browning (foxing) I am quite dubious because good lighting or camera flash can hide much of it, meaning it could be worse than expected.