The extension conjecture for the Calabi homomorphism on the disk

Let $Ham(D^2,\partial D^2)$ be the group of Hamiltonian diffeomorphisms of the disk preserving the boundary condition, and let $Hameo(D^2,\partial D^2)$ be its Hamiltonian-homeomorphism analogue. The Calabi homomorphism is

$$\operatorname{Cal}:Ham(D^2,\partial D^2)\to\mathbb R.$$

Calabi extension conjecture. The Calabi homomorphism extends to a homomorphism

$$\overline{\operatorname{Cal}}:Hameo(D^2,\partial D^2)\to\mathbb R$$

that is continuous in the Hamiltonian topology. Such an extension would follow from the preceding descent conjecture and the Calabi property of $\overline\mu^{path}$; the source gives no resolution.