As far as I know this is still an open problem. It is listed as an open problem in the paper Which crossing-number is it anyway? by Pach and Tóth, and also in the introduction of this more recent survey by Schaefer. Indeed, Schaefer notes that some authors define the crossing number as the pair crossing number, which can lead to immense confusion, and sometimes incorrect arguments. Even though the two parameters may be equal for all graphs, one cannot transpose the notions while writing a proof.
