Perhaps I don't understand gyroscopes enough, but is the Z-axis in line with the gravitational pull? When you start going around the curve of the Earth the Z-axis is just gonna match the pull of gravity. Nothing will change.
I agree with Jroa that there is nothing in the experiment to produce evidence for either side.
Let's imagine to be on a ship. We can draw three axes: a fore and aft axis, or if you prefer a prow-to-stern axis; let's name this the
X axis (but letters don't matter too much);
a port-to-starboard or a left-right axis, let's name this the
Y axis;
a vertical axis, let's name it the
Z axis.
Now please let's distinguish the difference between
rotate around an axis and
changing direction of an axis. Here i am speaking about rotating around an axis.
First case: rotation around the
X (prow-stern) axis: irrelevant because it depends from the rolling of the ship, and ships can roll in a flat earth and in a round one either;
Second case: rotation around the
Z (upper-down) axis: irrelevant because it depends from the ship turning left or right, and ships can turn on flat earth and round either;
Third case: rotation around the
Y (left-right) axis:
relevant because on a flat earth it can't change (trip always planar), while on a spherical earth it surely changes (it has to be a rather long trip though).
AFAIK there is a similar orientation system based on gyroscopes on the planes.
EDIT: maybe the words
rolling and
roll aren't the right ones, perhaps i should have written
tilting and
tilt, but you got the picture.