Sharp two-dimensional Szegő limit conjecture on the sphere

Let $S^{2}$ be the two-sphere, let $\mu$ be the measure used in the definitions of the functionals $A_{n}$ and $B_{n}$, and let $A_{n}(\varphi)$ and $B_{n}(\varphi)$ be the corresponding functionals for $\varphi\in L^{2}(S^{2},\mathbb{R})$. Sharp two-dimensional Szegő limit conjecture. For any $\varphi\in L^{2}(S^{2},\mathbb{R})$,

$$A_{n}(\varphi)\leq 0,$$

or, equivalently,

$$B_{n}(\varphi)\leq \frac{1}{2}\int |\nabla\varphi|^{2}\,\mu +(n+1)\int \varphi\,\mu;$$

equality holds if and only if $\varphi$ is a constant function. This conjecture is motivated by the classical one-dimensional Szegő limit theorem and is presented as a sharp reformulation of the determinant inequality on the two-sphere. The supplied text does not state that it has been proved or disproved.