For brief, precise, and hopefully readable descriptions of the current state of the art, I suggest:
this illustrated introduction (section 1.1, 1.2, 1.3)
this list of applications (section 6)
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.
Update (11/11/2019): In the updated version of GPS2, we prove descent for general sectorial covers. This is in my view better than the assertion that the Fukaya category is a cosheaf over a skeleton. 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 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 will appear in my upcoming 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.)
Finally I want to clear up what seems to be a common confusion.
The cosheaf property along the Lagrangian skeleton, or equivalently the gluing of Fukaya categories of Liouville sectors, is sometimes mirror to the fact that the functor category of coherent sheaves carries colimits to colimits.
There is a different sort of local-to-global principle which the Fukaya category should satisfy, mirror to various descent statements in algebraic geometry; i.e. the fact that category of coherent sheaves is a sheaf in various topologies. This was proposed by Seidel, and established for Fukaya categories of noncompact surfaces by Lee. It is unknown to me whether there is an explicit general formulation of this principle.
Note that the former procedure is local over the Lagrangian skeleton, and the latter is local over e.g. the tropical skeleton. I do not know any actual relation between Lagrangian and tropical skeleta, although one can imagine that one can produce the former from some sort of combinatorial Morse work on the latter.
Similarly I do not know a general relation between the two sorts of locality, except that one can sometimes use the former to prove the latter. E.g., one can read this paper as using Lagrangian skeleton locality to offload the hard work of proving mirror-to-affine-locality to a competent sheaf theorist.