  Tsirelson's original inequality for the precession protocol, first introduced
for the harmonic oscillator but applicable to all uniformly-precessing systems,
serves as a monopartite test of quantumness. Given that the system is
undergoing a uniform precession, the signs of the positions of a classical
oscillator must satisfy that inequality, which is violated by certain quantum
states. We consider this inequality for measurements with finitely many
outcomes in a theory-independent manner. We derive a general bound which
depends only on the minimum positive and negative values of the spectrum of the
observable. Given any such two values, we construct a quantum observable that
saturates this bound. A notable example is the angular momentum of a spin-$3/2$
particle. We also relate our findings to the recently-introduced notion of
constrained conditional probabilities.
