Why believe Kontsevich cosheaf conjecture?

On the suggestion of DamienC, I'm converting my comments into an answer. I didn't do this before because I don't really know the answer to the deeper question of why the microlocal sheaf category is the correct thing to give the Fukaya category (and I mostly just say things which are either obvious or wild speculation).

Some naive reasons why it's reasonable to expect local information on the skeleton to know everything:

1. It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section.

2. Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectic topology of the completion should be determined by the germ of the manifold along the skeleton.

More speculatively:

3. In mirror symmetry, you could think of $mathbf{R}^n$ as a torus with very large radius (like how $mathbf{R}$ is a circle with very large radius). When taking the dual Lagrangian torus fibration, a non-compact $mathbf{R}^n$ fibre should therefore be dual to a point (circle with very small radius). You see examples of this in recent work by Lekili and Polishchuk (https://arxiv.org/abs/1705.06023), where they find mirrors to punctured surfaces which are nodal curves: the structure sheaf at the node is mirror to a wrapped Lagrangian brane which is a non-compact $mathbf{R}$ going off to the puncture. So what is mirror to the Lagrangian $mathbf{R}^n$-fibration of the cotangent bundle by cotangent fibres? The zero section itself. Quite how you figure out that the derived category should be replaced by the category of microlocal sheaves, I don't know.

In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra": https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf

Edit^2: The following comments are misguided but I'll leave them here to give context to John Pardon's clarifying (and very illuminating) comments below.

Edit: I should add that the conjecture isn't going to be true if you allow arbitrary skeleta, only skeleta with mild (arboreal) singularities. In a talk by Daniel Alvarez-Gavela the other week, he mentioned the following example to see why this is true.

Take a Legendrian knot in the sphere (boundary of the ball) and attach a Weinstein handle along it. For a suitable choices of Liouville vector field, the skeleton of the resulting handlebody is just the core of the handle union the cone on the Legendrian. Topologically this is just homeomorphic to a sphere (but it is very singular at the cone point). The skeleton therefore doesn't depend on the knot, but the Fukaya category of the handlebody certainly does. You need to "arborealise" the singularity before you get a skeleton to which the Kontsevich conjecture can possibly apply.