Suppose that M is an $n\times n$ upper triangular matrix, with 1's on the diagonal, for which $-1\leq m_{ij}\leq 0$ for all $i<j$. Let N be the matrix $M^{-1}M^T$. Prove that the complex eigenvalues of N have absolute value 1.
I have shown this for $n\leq 4$. In general, the lower bound on $m_{ij}$ cannot be dropped completely, as I have found an 8 x 8 counterexample in which one of the $m_{ij}$ is -5 and the others 0 or -1. However, extensive attempts with Mathematica to contradict the stated conjecture have proved unsuccessful.
This problem has a long history and is related to Coxeter groups. For instance, R.Howlett (Bull.London Math.Soc. 14 (1982), 137-141) gives a simple argument that it is true if $M+M^T$ is positive definite.