Refining the constant in a maximum principle for the Bergman space

Let $A^2(\mathbb{D} )$ be the Bergman space over the open unit disk $\mathbb{D}$ in the complex plane. Korenblum conjectured that there is an absolute constant $c,~0<c<1$, such that whenever $\vert f(z)\vert\leq \vert g(z)\vert$ ( $f,g\in A^2(\mathbb{D} )$) in the annulus $c<\vert z\vert<1$, then $\Vert f\Vert\leq \Vert g\Vert$. In this note we give an example to show that $c<0.69472.$