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This completes a long-standing line of inquiry into sharp eigenfunction bounds for the Hermite operator, developed through the works of Thangavelu, Karadzhov, Koch--Tataru, and others. In higher dimensions \\(d\\ge 3\\), the corresponding endpoint estimates \\[\n  L^2(\\mathbb{R}^d)\\to L^{\\frac{2(d+3)}{d+1}}(\\mathbb{R}^d) \\] were recently established by the present authors. 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