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We verify the relative Langlands duality conjecture proposed by Ben-Zvi,Sakellaridis, Venkatesh for the hyperspherical Hamiltonian variety$T^*(\operatorname{Sp}_{2n}\backslash \operatorname{GL}_{2n+1})$. We providenumerical (over number fields and function fields) and geometric (in the\'{e}tale setting) evidence that its dual Hamiltonian variety should be$T^*(\operatorname{GL}_n \times \operatorname{GL}_{n+1} \backslash\operatorname{GL}_{2n+1})$ as is predicted by Ben-Zvi, Sakellaridis, Venkatesh.

On the relative Langlands duality for $\operatorname{Sp}_{2n} \backslash \operatorname{GL}_{2n+1}$ (with an appendix by Zeyu Wang)