Exponential PSD rank of positively shifted matching matrices

For every fixed $0<\rho <1$, the matrix indexed by odd vertex sets U and perfect matchings M of K${}_n$, with entries $|M\cap \delta (U)|-1+\rho$, has real positive semidefinite rank $2^{\Omega (n)}$ as even n tends to infinity. Here $\delta (U)$ is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.