since the earth is not perfectly in proportion (Nothing is perfectly in proportion in this reality). This is theoretically possible, but proportionally impossible <just a warning)
if there was a perfect sphere with perfect weight distribution - it would theoretically be possible for a particle to be suspended as long as it is at the perfect center? Thats if there is an equal amount of pull from all sides. Nothing we know of (especially the earth - has an equal weight on ALL sides)
You can actually take it one step further. Any particle at any point inside a uniform spherical shell is attracted equally in every direction by the shell. There’s numerous proofs for this.
-the earth is ellipsoid but you could put the particle in the horizontal+vertical center and the pulls on it should still be symmetric
-unless the center of mass isn’t exactly in the center
-wherever the center of mass is, i think if you put the particle there (with empty space around it) it would stay suspended
-the center of mass might not stay constant given magma flows, etc.
-if you get the particle close to the center it might take a long time for it to gain a noticeable amount of speed in any direction.
-due to quantum uncertainty it would be impossible to get the particle perfectly in the center