The twisted Fukaya equivalence for normal crossings surfaces

Let $X$ be a normal crossings surface with graph-like singular locus and orientable dual intersection complex, arising as the zero locus of a section $s$ of a line bundle $L$ on a regular $3$-fold $Y$. Let $\Sigma$ be the Riemann surface dual to the singular locus of $X$, and let $M$ be the symplectic $4$-manifold obtained from $P(K_\Sigma)$ by the described regluing. Twisted Fukaya equivalence. There is a $k$-linear equivalence

$$\operatorname{DSing}(X)\cong \mathrm{RFuk}^{\mathrm{gr}}(M).$$

Moreover, $\mathrm{RFuk}^{\mathrm{gr}}(M)$ has an autoequivalence $\Lambda$ and a $\Lambda$-twisted $2$-periodic structure, under which tensoring by $L$ corresponds to $\Lambda$ and the two twisted periodic structures agree. This is a conjectural mirror-symmetry description for the singularity category.