If a hadamard matrix of order 24 has character exactly 2, its transpose is known

In [1], the order 24 Hadamard matrices of character at least 3 was classified. In this paper, we show that those with character exactly 2 have already been classified since the transpose character must be at least 3. It is still not known if there are inequivalent matrices of this order with both character and transpose character 1.