Uniqueness of the complex $4\times 8$ equiangular tight frame

A complex $4\times8$ ETF is an equiangular tight frame represented by a matrix with four rows and eight complex unit-norm columns. Two ETFs are switching equivalent when they are identified under the switching operations used for ETF signature matrices.

$4\times8$ uniqueness conjecture. There exists a unique complex $4\times8$ ETF up to switching equivalence.

The paper reports a failed initial Gröbner-basis attempt and numerical evidence supporting the claim, including trials whose rounded signature matrices produced the same ETF. The conjecture is not proved in the source.