From that post:I gave here a very concrete example of a joint position and momentum measurement of quantum particles.
So the observable being measured is neither position nor momentum; it's a POVM which is a different operator from either of those. Certainly it is not a Hermitian operator describing "measure position and momentum simultaneously", since no such Hermitian operator exists (the product of the position and momentum operators is not Hermitian). This POVM can be described in ordinary language as "a measurement of both position and momentum" only because "measurement" is here being used in a different sense than the sense in which it is being used in the OP's question.Note that we measure both position and momentum, which is not covered by Born's rule.
But it is described by a POVM with an operator for each of the ##w^L## possible signal patterns.