I was reading that Greece and Turkey went head to head on oil exploration in the Aegean, but both of them backed down and agreed not to explore (in the late 80's or early 90's).
Probably that was for the sea between Greece and Turkey. If the two countries could not agree for an undersea border, then any oil exploitation would be difficult.
However in this case the sea in question is not between Cyprus and Turkey, but between Cyprus and Lebanon, and Cyprus and Egypt. Cyprus has already agreed for the undersea border with those two countries and it is really not Turkey's business.
However what is a problem is the plan of Turkey to exploit oil between Cyprus and Turkey. For that to happen Republic of Cyprus and Turkey should agree for an undersea border between them. If that doesn't happen any exploitation by Turkey will be illegal.