Gerstenhaber conjecture for free loop space I- Is the following statement still a conjecture see this article ?
Conjecture (?)
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild cohomology of cochain complexes associated to $M$ is isomorphic as a Gerstenhaber algebra to $H_{\ast+d}(LM)$ the $d$-shifted homology of the free loop space on $M$.
II- What are the recent advances in solving the conjecture ? 
III- Is there a short proof of the following fact: $HH^{\ast}(C^{\ast}(M))$ is isomorphic to $H_{\ast+d}(LM)$ as a graded vector space ?
 A: I. To my knowledge over any field this conjecture is still open. 
II. Over a field of characteristic zero, this conjecture is true, and we have more, we have a BV-isomorphism:
Félix, Yves; Thomas, Jean-Claude
"Rational BV-algebra in string topology." 
Bull. Soc. Math. France 136 (2008), no. 2, 311–327.
A: For III:
First show that the Hochschild homology of the DGA of cochains on $M$ is quasi-isomorphic to cochains on $LM$. To do this you can use Jones argument using a cosimplicial model for $LM$ or over the rationals one can construct an explicit map through Chen iterated integrals. 
Now the "Poincaré duality step": relate Hochschild homology to Hochschild cohomology.  To do this one may construct a lift of the iterated integrals map mentioned above by choosing a Thom form supported near the diagonal in $M \times M$. The details of this approach can be found on Merkulov's De Rham string topology. 
There are other ways of achieving this last Poincare duality step by using finite dimensional models for the cochains on $M$: either strict Frobenius algebras  or cyclic $A_{\infty}$ algebras. 
