Why are square planar coordination compounds with four different ligands optically inactive?

Question

Why do square planar coordination compounds of type $\ce{[Mabcd]}$ not show optical activity, although they contain 4 different ligands (i.e. chiral central metal atom)?

Answer 1

Even though the central atom has 4 different ligands this does not necessarily mean that the compound is optically active. The condition for that rather the following (from Wikipedia):

A molecule is achiral (not chiral) when an improper rotation, that is a combination of a rotation and a reflection in a plane, perpendicular to the axis of rotation, results in the same molecule[...] For tetrahedral molecules, the molecule is chiral if all four substituents are different.

So for 4-coordinated carbon atoms which are tetrahedral having four "ligands" actually establishes optical activity. But in the case of a square planar metal complex you always have a mirror plane, namely the one made up by the central atom and the ligands:

And as a result you don't get optical activity. But it might be possible that some vibrational modes destroy the mirror plane symmetry element of the metal complex and this might lead to some optical activity depending on how well this mode excited.