The preparation of quantum many-body systems faces the difficulty that in a realistic scenario only few control parameters of the system may be accessible. In this context, an interesting connection between the energy fluctuations during state preparation and its geometric length as measured by the Fubini-Study metric was discussed in [5]. An inspiring conjecture lower bounding the energy fluctuations by the minimal geometric length of all accessible state preparation protocols was put forward together with supporting examples and numerical evidence. However, we here show that the conjecture does not hold but can be violated if the accessible parameter space has extrinsic curvature, when embedded into the space of all dynamically accessible states. We illustrate this by a number of generic qubit, qutrit and harmonic oscillator systems.