Conjecture on rank-two partial isometries being unitarily equivalent to complex symmetric matrices

Let $T$ be a partial isometry on a $4$-dimensional complex Hilbert space, and suppose that $\operatorname{Rank} T=2$. A matrix $T$ is unitarily equivalent to a complex symmetric matrix if there is a unitary matrix $U$ such that $U^*TU$ is equal to its transpose. Rank-two partial-isometry conjecture. Every rank-two $4\times 4$ partial isometry is unitarily equivalent to a complex symmetric matrix. The cases of rank $0$, $1$, $3$, and $4$ are known, while the rank-two case was unresolved in the source and supported there only by computational testing of $100{,}000$ random examples.