This is a generalized version of Does a non-singular matrix have a large minor with disjoint rows and columns and full rank?

Let $A$ be an $n$-by-$n$ antisymmetric matrix of rank $r\geq \epsilon n$. Is there a minor of $A$ with disjoint row and column indices $I,J\subset \{1,2,\dotsc,n\}$ and rank $k\geq \lfloor r/1000\rfloor$?

(Some argument involving a Pfaffian might work, but the matter does not seem self-evident (to me).)