In the case of a Russo-NATO war in the Baltics, is there any chance that Turkey opens up a second front in the Caucuses?

Turkey has the second largest military in NATO, and operates a number of heavy units, unlike many NATO members. Furthermore, Georgia, which borders both Turkey and Russia, is very hostile toward the Russians.

Is there any chance that Turkish forces, possibly bolstered by American troops, gain passage through Georgia and begin to launch an offensive into southern Russia? With Russian forces heavily engaged in the Balkans, needing to keep significant forces on the Ukrainian border to secure Crimea, Donbass, and guard against a Ukrainian attack, the southern border might be undermanned. Of course, Turkey is heavily engaged in Syria and against the PKK, and the Turkish armed forces are in a precarious political position.

Is there any chance of a Turko-Georgian operation into Southern Russia? Past that, is there a chance that Turkey would close the Straits to Russian shipping, and possibly let an American carrier into the Black Sea? On the other hand, with Turkey and Russia rapidly rapproaching, would the Turks even declare war on Russia as a member of NATO, and would they begin to conduct operations against Russia's Black Sea fleet and ports?