Hogancamp's Hamiltonian-action conjecture for link homology

Let $L$ be an arbitrary link, let $\mathrm{HY}(L)$ denote its homology, and let $\mathcal{E}_k$ and $\mathcal{F}_k$ be the operators acting on it. Let $\mathcal{H}_2$ be the Lie algebra of Hamiltonian vector fields on the plane, and let $\mathcal{H}_2^{\ge 2}$ be the subalgebra spanned by the vector fields $v_{x^ay^b}$ with $a+b\ge 2$. For a knot $L$, let $\overline{\mathrm{HHH}}(L)$ denote its reduced HOMFLY-PT homology. Hamiltonian-action conjecture. Given an arbitrary link $L$, there is an action of the Lie algebra $\mathcal{H}_2$ on $\mathrm{HY}(L)$ extending the action of $\mathcal{E}_k,\mathcal{F}_k$. If $L$ is a knot then there is an action of $\mathcal{H}_2^{\ge 2}$ on $\overline{\mathrm{HHH}}(L)$. This proposes that the Hamiltonian symmetry established for $(n,n+1)$ torus knots extends to all links, with the reduced version for knots. The source gives no further resolution of the conjecture.