Auroux's mirror correspondence conjecture for restriction, wrapping, and lifting functors

Let $X$, $D$, and $Z$ be the algebraic-geometric mirror objects, with inclusion $i_X:X\to Z$, projection $\pi_X:Z\to X$, and inclusion and projection

$$j_D:K_X|_D\to Z,\qquad p_D:K_X|_D\to D.$$

Let $q:\operatorname{Coh}(Z)\to\operatorname{Sing}(Z)$ be the quotient functor, and let $\operatorname{Sing}(Z)\to\operatorname{Coh}(D)$ be the Knörrer periodicity equivalence. The Fukaya-side functors are the restriction functor $\rho$, the wrapping-around-$H$ functor $\alpha_0$, the wrapping-at-infinity functor $\alpha_\infty$, and the lifting functor $j$.

Auroux's mirror correspondence conjecture. Under homological mirror symmetry, $\rho$ corresponds to the composite of $q$ with the Knörrer periodicity equivalence; $\alpha_0$ corresponds to $i_{X*}$; $\alpha_\infty$ corresponds to $\pi_X^*$; and $j$ corresponds to

$$j_{D*}p_D^*(K_X|_D\otimes\mathord\cdot\,).$$

These conjectures identify natural Fukaya-category functors with quotient, pushforward, pullback, and tensor-pushforward functors in algebraic geometry. The source attributes the correspondence to Auroux and discusses the individual claims in the indicated sections; the supplied text gives no evidence that they have been resolved.