Liu's exact norm conjecture for the Bergman projection

Let $P$ be the Bergman projection on the unit disk. For $1<p<\infty$, let $q:=p/(p-1)$ be the conjugate exponent and let $\|P\|_p$ denote the operator norm of $P$ on $L^p(\mathbb{D})$. Liu's conjecture. For every $1<p<\infty$,

$$\|P\|_p=\Gamma(2/p)\Gamma(2/q).$$

The conjecture is motivated by Liu's lower bound, which disproved Dostanić's earlier conjecture. The source does not report a resolution of Liu's conjecture.