Measuring the fluctuations of work in coherent quantum systems is notoriously problematic. Aiming to reveal the ultimate source of these problems, we demand of work measurement schemes the sheer minimum and see if those demands can be met at all. We require ($mathfrak{A}$) energy conservation for arbitrary initial states of the system and ($mathfrak{B}$) the Jarzynski equality for thermal initial states. By energy conservation we mean that the average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Requirement $mathfrak{B}$ encapsulates the second law of thermodynamics and the quantum--classical correspondence principle. We prove that work measurement schemes that do not depend on the systems initial state satisfy $mathfrak{B}$ if and only if they coincide with the famous two-point measurement scheme, thereby establishing that state-independent schemes cannot simultaneously satisfy $mathfrak{A}$ and $mathfrak{B}$. Expanding to the realm of state-dependent schemes allows for more compatibility between $mathfrak{A}$ and $mathfrak{B}$. However, merely requiring the state-dependence to be continuous still effectively excludes the coexistence of $mathfrak{A}$ and $mathfrak{B}$, leaving the theoretical possibility open for only a narrow class of exotic schemes.
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