Strong dualizability conjecture for Borel equivalence relations

Let $X$ be a standard Borel space, and let $E$ be a Borel equivalence relation on $X$. Strong dualizability conjecture. Every such Borel equivalence relation is strongly dualizable. This would establish strong duality in full generality for Borel equivalence relations; the source reports that no counterexample to strong dualizability is known, while suggesting that a pathological example may be possible outside this setting.