“...Turkey might well escalate again in response to the Egyptian Army entering Libya. If those two end up in direct conflict, it could be a major war....”
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It wouldn’t be much of a “major” war. Turkey is incapable of projecting sufficient military power for any significant length of time.
Turkey is unlikely to deploy a bunch of their own Regular Army and heavy armor in a rapid fashion for a Desert Storm-style high intensity war, but they could probably send a thousand or two jihadis per month relatively easily, with all the small arms ammo they could shoot, as well as anti-armor missiles.
If the Qataris pick up the bill for salaries, food, vehicles, fuel, medical and such; Turkey has one of the planet’s top ten arms industries, to supply the bang.
The whole team that ran the Syrian Civil War on the Sunni jihadi side, could shift the same operation to Libya. They have already done so to a very significant degree.
After a year or two of that, the body count would be climbing the charts, just like it did in Syria (which ended up being the biggest refugee movement in fifty years).