There can no longer be any doubt, Mihawk is a Conqueror. Chapter 1033 makes it unambigously clear.
To break down what happened. Zoro unleashes his Haoshoku Haki and knocks down fodder.
King notices it and comments on Zoro’s “kingly ambitions”.
Immediately, after, we’re shown what those “kingly ambitions” of Zoro are.
He thinks back to his promise to Luffy to become the World’s Strongest Swordsman.
And then explicitly states that King is right, as he made such promises to his captain and his best friend Kuina.
Mihawk is the current World’s Strongest Swordsman. To aspire to be the World’s Strongest Swordsman is to seek to reign above all swordsmen, to become a King. If Zoro who merely holds that ambition is a Conqueror, then how much more so the current occupant of the title?
Mihawk who reigns at the top and is waiting for Zoro. He’s the very one that Zoro seeks to surpass.
And it goes without saying that if Zoro is capable of Haoshoku Haki coating then Mihawk also would be.
But more than just being any ordinary Conqueror, Mihawk is probably one of the greatest of Conquerors.
Zoro awakened as a Conqueror in order to pass Enma’s test.
How much more so Mihawk who not only mastered the “finest blade” in the world but (most likely) also forged it into Kokuto.