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:
RoosRoos, Bero, B. On Bobkov's approximate de Finetti representation via approximation of permanents of complex rectangular matrices. ProcOn Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices, Proc. AmerAm. Math. Soc. 143 (2015), noNo. 4, 1785--1796 (2015). ZBL1345.28002.