Fukaya category conjecture for compatible tangential structures

Let $(\Theta,\Phi)$ be a tangential pair, meaning based spaces equipped with maps fitting into a commutative diagram over $BO \to BU$. Let $X$ admit a $\Phi$-structure, set $A:= \operatorname{Thom}(\Omega^2\Phi \to BU \times \mathbb{Z})$, and let $R$ be the Thom spectrum of the stable vector bundle over $\Omega F$, where $F$ is the homotopy fibre of $\Theta \to \Phi$. Compatible tangential Fukaya-category conjecture. There exists a Fukaya category $\mathcal{F}(X;A,R)$ whose objects are exact Lagrangians $L$ equipped with a compatible $\Theta$-structure: the diagram $L \to \Theta \to BO$ over $X \to \Phi \to BU$ commutes, with the horizontal composites classifying $TL$ and $TX$. The construction would extend Lagrangian Floer theory to tangential pairs, with coefficients controlled by the $A$-algebra $R$; existence of this Fukaya category is not established in the supplied text.