Homogeneous-coordinate-ring conjecture for the mirror filtration

Let $U\subset Y$ be an snc partial minimal model, let $W$ be an effective ample divisor supported on the boundary, and let $\widetilde A_{\theta}$ be the graded subring obtained from the filtration defined by the tropical orders along the boundary components. Homogeneous-coordinate-ring conjecture. One fibre of $\widetilde A_{\theta}$ is the homogeneous coordinate ring of $(Y,\mathcal O(W))$.

This conjecture proposes that the mirror-algebra filtration recovers projective geometry of the partial minimal model. The supplied text gives no evidence of resolution.