Homological mirror symmetry conjecture for the Barlow surface

Let $S$ be a Barlow surface equipped with a generic Kähler form. An orthogonal decomposition of its Fukaya category is a decomposition into mutually orthogonal full subcategories, and $\operatorname{MF}(\mathbb{C}(x,y)/x^2+y^2)$ denotes the indicated matrix factorization category. The Barlow-surface HMS conjecture. The Fukaya category of $S$ has an orthogonal decomposition into a phantom category and eleven matrix factorization categories of type

$$\operatorname{MF}(\mathbb{C}(x,y)/x^2+y^2).$$

This is a proposed extension of homological mirror symmetry to varieties of general type, predicting a Fukaya-category counterpart to the phantom decomposition found in the derived category. The source gives no resolution.