[tengbretson]

The plant will zero out the potential, presumably because there is a small current through the plant itself. In the presence of a nearby antenna/grounding rod the current through the plant would be significantly lower, since there is a less resistive path to ground.

In the case of my hypothesis, the mechanism I'm describing would likely be somehow tied to the plant's perception of that current.

[hwillis]

It's in the range of femtoamps. Current in general is not really very directly measurable, particularly for biology. Instead you do things like measuring voltage across something or magnetic fields.