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, Bero, On Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices, Proc. Am. Math. Soc. 143, No. 4, 1785-1796 (2015). ZBL1345.28002.
