Almost Hadamard conjecture in the complex setting

An almost Hadamard matrix in the complex sense is the notion introduced in the cited work, while a complex Hadamard matrix is a unitary matrix whose entries have equal modulus after the usual rescaling. Almost Hadamard conjecture. Any almost Hadamard matrix, taken in a complex sense, must be actually a complex Hadamard matrix. Equivalently in the paper's discussion, every local maximizer of the $1$-norm on the unitary group $U_N$ should be a global maximizer. The source cites geometric and defect computations as evidence but gives no proof.