A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely generated $\mathbb{C}$-algebra.

On the other hand P.M.H. Wilson (using a construction of Zariski) gave an example of a compact complex manifold $X$ with $R(X)$ not finitely generated. However his manifold $X$ is not Kähler.

Does anyone know an example of a compact Kähler manifold $X$ with $R(X)$ not finitely generated? Or is this an open problem?