Eliashberg–Hofer conjecture on contact homology invariance

Let $(M^{2n-1},\xi)$ be a contact manifold with contact form $\alpha$ for $\xi$, and let $C_*$ be the chain complex generated by closed Reeb orbits, with differential $d$ defined by counts of pseudoholomorphic cylinders. Assume that

$$C_k=0\quad\text{for }k=-1,0,1.$$

Eliashberg–Hofer conjecture. The differential satisfies $d^2=0$, and the homology $H_*(C_*,d)$ is independent of the contact form $\alpha$ for $\xi$ and of the complex structure $J$.

This asserts that, under the stated vanishing condition, the resulting contact homology is well defined and depends only on the contact structure rather than on the auxiliary contact form or almost-complex structure. The supplied source attributes the claim to Eliashberg and Hofer, but gives no resolution evidence.