Is the closure of $[\sigma_1^2,\sigma_2^2]$ in $B_3$ equal to the Borromean rings?

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Is the closure of $[\sigma_1^2,\sigma_2^2]\in B_3$ (the braid group with $3$ strings) equal to the Borromean rings? If yes, is there any simple proof?

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Yes.

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Arguably the simplest proof.