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Does the $Z$ boson interact with gravity? I understand it has a measurable mass. I'm instead looking for an experiment that says, unequivocally, it interacts with gravity (as opposed to, e.g., having inertial mass).

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To the best of my knowledge, we hardly have a solid experimental understanding of gravity at the atomic scale. Hence, I doubt there is any experiment that can claim to have checked that the $Z$ boson interacts gravitationally.

Nevertheless, general relativity—our best understanding of gravity nowadays—describes gravity as spacetime curvature, which means it affects everything. It doesn't even matter if the matter has mass. Light, for example, is also subject to gravity. Assuming GR, or some adapted version of it, holds up to the electroweak scale, then the $Z$ boson surely interacts gravitationally. I know of no serious reason to doubt whether the main principles of GR hold to such "mild" scales, although they might fail considerably near the Planck scale.

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It's very unlikely any such experiment will ever be done.

The Z boson is produced only in high energy collisions where the particle is moving at close to the speed of light, and its lifetime is about $3 \times 10^{−25}$ seconds. Any direct measurement of the gravitational deflection of its trajectory would be impossible as it would be immeasurably small.

In general it is very difficult to measure the interaction of fundamental particles with gravity as the interaction is so weak compared to the other forces. For example we don't even know if antimatter responds to gravity in the same way matter does.

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