Second-order stationary-point conjecture for quantum compiling

Let $V_{\mathsf{ct}}(\cdot)$ be the circuit-to-unitary map, let $U$ be a target unitary, and define

$$f_{\mathsf{ct}}(\cdot)=\frac{1}{2}\|V_{\mathsf{ct}}(\cdot)-U\|_{\mathrm{F}}^2.$$

A stationary point ${\mathbold{\theta}}$ is a point where the gradient vanishes, and it is second-order stationary when the Hessian of $f_{\mathsf{ct}}$ is positive semidefinite there. Second-order stationary-point conjecture. If $V_{\mathsf{ct}}(\cdot)$ is surjective and the Hessian of $f_{\mathsf{ct}}$ is positive semidefinite at a stationary point ${\mathbold{\theta}}$, then

$$V_{\mathsf{ct}}({\mathbold{\theta}})^{\dagger}U=e^{i\alpha}I$$

for some $\alpha\in[0,2\pi)$. This would imply that every second-order stationary point is a global minimum up to a global phase; the supplied text reports extensive experiments but gives no proof or resolution.