Proposition: Local parity attains the unrestricted value

The effect

is implementable by measuring each qubit in the $X$-basis and accepting the even-parity outcomes. It has bias $1/2$. Thus the exponential bound in Theorem thm:productgap is not a limitation on general product POVMs, separable effects, or local measurements with classical post-processing.

Proof. The even-parity effect is the sum of the corresponding tensor-product $X$-basis projectors, so it has the stated local implementation. The coherent GHZ state has $X^{\otimes c}$-expectation one. Its record-dephased mixture has expectation zero. Their acceptance probabilities are therefore one and $1/2$.