Minimum maximum-entry norm of orthogonal matrices in dimensions five, six, and seven

Determine whether \(u(5)=6/11\), \(u(6)=1/\sqrt{5}\), and \(u(7)=1/7+3\sqrt{2}/14\), where \(u(n)=\min_{M\in O(n)}\max_{i,j}|M_{ij}|\), and establish the corresponding minimizers where they are specified.

Background

The paper studies the minimum possible maximum absolute entry of an orthogonal matrix, connecting the quantity to Hadamard matrices. After proving $u(3)=2/3$, the authors propose explicit conjectured values for dimensions five, six, and seven. These claims are not proved in the paper.