I was informed by a colleague that there is also a solution to the Pfaffian formulation of the problem, in Lemma 2.1 of the following article:
Roos, B. On Bobkov's approximate de Finetti representation via approximation of permanents of complex rectangular matrices. Proc. Amer. Math. Soc. 143 (2015), no. 4, 1785--1796.