The noncommutative divisor deformation conjecture for the fibrewise compactification

Let $\EuScript A$ be the Fukaya $A_\infty$-algebra of the Lefschetz fibration and $\EuScript B_{q,\bar b|M}$ the category associated to the fibrewise compactification. Suppose that the deformation $\EuScript A_{q,\bar b}$ is trivial. Noncommutative divisor deformation conjecture. The category $\EuScript B_{q,\bar b|M}$ has the structure of a deformation of noncommutative divisors on $\EuScript A$, whose leading-order term is the image of the deformation of the canonical natural transformation under the trivialization isomorphism. This conjecture extends the noncommutative-divisor structure established for the undeformed fibre category to fibrewise compactifications.