A comment re. Jonny's nice answer: there was indeed a time when that was the envisioned strategy of proof. However our present approach does not require the arborealization. Because: now we know that the Fukaya and microlocal categories associated to any (singular) Legendrian in a cosphere bundle agree.
In GPS2, we prove descent for general sectorial covers. In particular, one could deduce the existence of said cosheaf by an (unwritten) purely geometric argument that an open cover of the skeleton lifts to a sectorial cover of the symplectic manifold. However, the notion of sectorial cover is more flexible, as it is not tied to a particular skeleton; in a certain sense it captures all skeleta at once.
The real virtue of the cosheaf on the skeleton is that it is not some arbitrary cosheaf of categories, but in fact is the Kashiwara-Schapira stack of microlocal sheaf theory. (Historical note: this assertion goes beyond the original conjecture of Kontsevich, and is probably best attributed to Nadler.) As mentioned above, GPS3 is the required local
calculation. In principle, one could use GPS2 to glue, but actually there is a short-cut for most Weinstein manifolds of interest in mirror symmetry and geometric representation theory. The reason is that these manifolds typically have trivial stable symplectic normal bundle, so one can use the embedding trick without messing about with descent from a Lagrangian Grassmannian bundle. This reduces the problem to the cotangent comparison of GPS3, provided one has certain full faithfulness of embedding results in both partially wrapped Floer theory and in microlocal sheaf theory.
Full faithful embeddings come from a certain doubling construction, which was originally introduced in sheaf theory by Guillermou, it is described in e.g. section 11.4 of his omnibus. Guillermou assumes there that the relevant skeleton is smooth, but similar ideas give a construction in general and appear in Sec. 6 of my paper with Nadler. On the wrapped Floer theory side the construction is Example 8.6 in GPS2. Combining the above one learns that for a Weinstein manifold with stably trivial symplectic normal bundle, the wrapped Fukaya category is equivalent to the category of microlocal sheaves on any Lagrangian skeleton.
(Most likely the `embedding trick' also works to establish the result in the general case, but this requires a bit more complicated argument.)