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We prove that any free ergodic probability measure preserving (pmp) action $\\tilde B_n\\curvearrowright (X,\\mu)$ is W$^*$-superrigid in the following sense: if $L^{\\infty}(X)\\rtimes\\tilde B_n\\cong L^{\\infty}(Y)\\rtimes\\Lambda$, for an arbitrary free ergodic pmp action $\\Lambda\\curvearrowright (Y,\\nu)$, then the actions $\\tilde B_n\\curvearrowright X,\\Lambda\\curvearrowright Y$ are stably (or, virtually) conjugate. 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