Classification conjecture for full-dimensional thick te-interlaces

A thick te-interlace is a totally equimodular interlace satisfying the paper's thickness condition; a matrix is full-dimensional when it has the full dimension specified by that condition. Two matrices are equivalent up to resigning and permuting rows and columns.

Classification conjecture. There are only two full-dimensional thick te-interlaces, up to resigning or permuting rows and columns:

$$\left[\begin{array}{rrrr} 1&1&1&1\\ 1&-1&-1&1\\ 1&-1&1&-1\\ 1&1&-1&-1 \end{array}\right] \quad\text{and}\quad \left[\begin{array}{rrrrrr} 1&1&1&1&1&1\\ 1&1&1&1&-1&-1\\ 1&1&1&-1&-1&1\\ 1&1&-1&-1&1&1\\ 1&-1&-1&1&1&1\\ 1&-1&1&1&1&-1 \end{array}\right].$$

The conjecture is motivated by a brute-force computer enumeration showing that there are no thick te-interlaces of size $8$; the classification of all full-dimensional thick te-interlaces remains open.