Why would it? The only thing needed to get rid of plastic is destroying it's key advantage over other materials (price) either by force or will.
I am very convinced that science will bring us a better material e.g. by looking at nature if we are "forced" to do so, this can be through market force, e.g. oil becoming just too expensive. I'm thinking long term here.
Landfill is the only solution that is guaranteed to fail. We're living on a quasi-infinite planet with a population that is to this date increasing. It will be easier to get new resources onto the planet than it is to get them away from the planet and stuff piling up indefinitely is clearly infeasable.
Or… we can trust Exxon (ffs) to build a recycling plant of the future that somehow is green also.