It takes a while for various other industries to shift away from burning oil and gas, but when that happens the oil industry will be totally screwed.
I'm not so sure that they'd be necessarily screwed even then, I think it will depend upon what direction plastic demand and plastic production goes in. The majority of plastics still need to be made from petrochemicals, and the majority of plastics have to be virgin simply due to the inherent limitations on their recyclability.
Sure, the industry won't be as large as today but unless we see bioplastics completely replace petrochemical plastics or simply see plastics completely abandoned (that'll never happen, plastics are simply too useful to ditch entirely.) It will still exist in some form simply because it will be necessary for plastic production.
Same reason why oil prices keep going up even as demand drops, the price of oil is largely determined by OPEC+ and they will cut production until prices rise to where they want them, and the more they have to cut production the higher the prices have to be to offset their fixed costs. More oil = more total revenue = lower price floor for profits.