If by true free will you mean ability to do anything, then of course we don't have that. We are bound in a great many ways. We could be bound in infinite ways minus one and the one would be enough to give us will. Down this rabbit hole lies an understanding of a certain type of presentism that reveals how God can both give us will and know everything. There is no future, in short (but I can elaborate if desired). Also qualitivity needs to be properly understood. Jeff, again, would struggle here, as he contended that all of reality can be distilled to quantitivity (mathematics! and maybe some Dawkins crystals?), which is knowable to not be the case.

As far as an experiment, that's beyond the scope into the mysterious yet again, though an experiment belies outside purpose, and it would be difficult to surmise what possibility of purpose would obtain for an ontologic fundamental in pursuing one...