It's expected that mesh convergence becomes problematic as the support approaches a singularity (zero area). As soon as the support area becomes too small to prevent gross local deformation it is too small. Your 4 UY constraints should have prevented ROTX and ROTY rigid body motion (assuming the support areas are large enough.) Even though you have a way of dealing with the unconstrained lateral motion, it would be better to prevent it as having a zero stiffness can create numerical error.
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