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By recent results of Greb\\'ik and Recke and d'Achille et al., this \"uniqueness threshold\" $p_u(\\lambda)$ tends to zero as the intensity $\\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\\mathbb{H}^{d_1}\\times\\dots\\times\\mathbb{H}^{d_k}$ with $k,d_1,\\dots,d_k\\geq 2$. 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