The inner-product characterization of equivalent frames

Let $F=\{f_i\}_{i=1}^k$ and $G=\{g_i\}_{i=1}^k$ be elements of ${\mathbb{S}}(n,k)$, the class of frames under consideration. For each pair of distinct frame vectors, consider the absolute inner products $|f_i^Tf_j|$ and $|g_i^Tg_j|$.

Equivalent-frame inner-product conjecture. $F$ and $G$ are equivalent if and only if

$$\{|f_i^Tf_j|, 1\leq i<j\leq k\}\text{ is a permutation of }\{|g_i^Tg_j|, 1\leq i<j\leq k\}.$$

This would give a general correctness guarantee for the Inner Product Algorithm, which tests equivalence by comparing the absolute pairwise inner products. The corresponding characterization is established in the paper for ${\mathbb{S}}(2,3)$, while the general case is presented as a conjecture.