Norm of the Hilbert matrix on Bergman spaces

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space $A^p$ into $A^p$ if and only if $2<p<\infty$. In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin ?\frac{2\pi }{p}}$, when $4\le p<\infty$, and it was also conjectured that

${6H6}_{A^p\to A^p}=\frac{\pi }{\sin ?\frac{2\pi }{p}},$

when $2<p<4$. In this paper we prove this conjecture.