Please God, no. Schoenborn has refused to crack down on leftist/dissenting priests in Austria, so why trust him with the whole Church??
I seem to remember Schoenborn drew a public rebuke from the Vatican over an incident in the last few years. If memory serves, he reversed a local Catholic tribunal’s decision to exclude an openly gay man from serving on its board AND invited that man and his partner to lunch at Schoenborn’s palatial home. Although B16 (Ratzinger) and Schoenborn are decades-long friends, B16 felt it important to rebuke his old friend. I doubt Schoenborn can get a majority of the conclave to vote for him in light of the rebuke, whatever one makes of merits of the underlying dispute.