Künneth conjecture for directed Fukaya categories of invertible polynomials

Fix a field $k$ and positive integers $p_0,\ldots,p_n>1$. For each $i$, let $W_{p_i}(X_i)\in\mathbb{C}[X_i]$ be a general polynomial of degree $p_i$, and set

$$W_{p_0,\ldots,p_n}(X_0,\ldots,X_n)=W_{p_0}(X_0)+\cdots+W_{p_n}(X_n).$$

Let $\dirFuk W_{p_0,\ldots,p_n}$ be the directed Fukaya category of the associated exact Lefschetz fibration, and let $\dirFuk W_{p_i}$ denote the directed Fukaya category associated with $W_{p_i}$. Künneth conjecture. There exists an equivalence

$$D^b\dirFuk W_{p_0,\ldots,p_n}\cong D^b(\dirFuk W_{p_0}\otimes\cdots\otimes\dirFuk W_{p_n})$$

of triangulated categories. This is a special case of a conjecture of Auroux, Katzarkov, and Orlov concerning weighted projective spaces and homological mirror symmetry; the statement compares the directed Fukaya category of a sum of one-variable polynomials with the derived category of the tensor product of the corresponding one-variable categories.