Uniqueness conjecture for homogeneous quasi-morphisms on Hamiltonian groups of complete Liouville manifolds

Let $M$ be a complete Liouville manifold, and consider its Hamiltonian group or its universal cover. A homogeneous quasi-morphism is a quasi-morphism $u$ satisfying $u(g^k)=k u(g)$ for every group element $g$ and integer $k$. The Calabi homomorphism is the standard homomorphism defined by integrating a compactly supported Hamiltonian over $M$. Uniqueness conjecture. The only homogeneous quasi-morphisms on the (universal cover of the) Hamiltonian group of $M$ are multiples of the Calabi homomorphism. The conjecture would classify homogeneous quasi-morphisms arising from Hamiltonian dynamics on complete Liouville manifolds. The preceding constructions show that several such functions exist on related groups or after restriction to Liouville subdomains, but the stated uniqueness remains unresolved here.