Refinement of the mean angle estimation in the Feyesh Toth problem

The Fejes Tóth problem about the maximum $E_{*}$ of the mean value of the sum of angles between lines in $\mathbb{R}^{3}$ with a common center is considered. L. Fejes Tóth suggested that $E_{*}=\frac{\pi}{3}=1.047\ldots$. This conjecture has not yet been proven. D. Bilyk and R.W. Matzke proved that $E_{*}\le 1.110\ldots$. We refine this estimate using an extremal problem of the Delsarte type: $E_{*}\le A_{*}<1.08326$. Using the dual problem $B_{*}$ we show that the solution of the $A_{*}$ problem does not allow us to prove the Fejes Tóth conjecture, since $1.05210<A_{*}$.