Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\Lambda \subset T^\infty M$, there is a category of "sheaves with singular support at infinity contained in $\Lambda$". (These are really complexes of sheaves, not individual sheaves, but I am trying to stay close to the language of my main source.)

I want to consider the case that $M$ is $\mathbb R^n$, which is equipped with a hyperplane arrangment. Each hyperplane in the arrangement is going to to appear in $\Lambda$, offset in the direction of a conormal. This means that the images of the hyperplanes in $T^\infty \mathbb R^n$ are not going to intersect. We choose the conormals consistently, so there is some tangent direction with respect to which they are all positive.

My question is the following: has anyone has figured out what the corresponding category of (complexes of) sheaves looks like in these examples coming from hyperplane arrangements?

The main paper I have been looking at where some of this is explained is arXiv:1512.08942 by Shende, Treumann, Williams, and Zaslow. They are interested in a somewhat different situation, where, in particular, $M$ is two-dimensional. They explain that, to understand the sheaves, it suffices to understand them on each of the regions of the arrangement (i.e., within each region, the stalks are canonically isomorphic). Further, in the case of a hyperplane arrangement in $\mathbb R^2$ with a single pair of crossing lines, the stalks of the four regions (cyclically, $S$, $E$, $N$, $W$) should be related by the fact that there is a commuting square:

$$ \begin{array}{ccc} S & \rightarrow & E\\ \downarrow &&\downarrow\\ W & \rightarrow & N\end{array}$$ and the total complex $S \rightarrow E \oplus W \rightarrow N$ should be acyclic.

Which region is which among the four has to do with the choices of the signs of the conormals.