Hamiltonian structure conjecture for antisymmetric L-functions

Let $\mathcal{L}$ be an L-function on $\mathcal{Z}$, and let $\hat{\mathcal{L}}_{F^{\mathrm{asym}}}$ be the contracted L-function defined by

$$\hat{\mathcal{L}}_{F^{\mathrm{asym}}}(\rho,u):=\inf_{\substack{j\in T_{\rho}\mathcal{W}:\,u=d\phi_{\rho}j}}\mathcal{L}_{F^{\mathrm{asym}}}(\rho,j).$$

Hamiltonian structure conjecture. There exist an energy $\mathcal{E}:\mathcal{Z}\to\mathbb{R}$ and a skew-symmetric operator $\mathbb{J}:\rho\mapsto(T^*_{\rho}\mathcal{Z}\to T_{\rho}\mathcal{Z})$ such that the zero-cost velocity of $\hat{\mathcal{L}}_{F^{\mathrm{asym}}}$ is

$$u^0(\rho)=\mathbb{J}(\rho)D\mathcal{E}(\rho).$$

The conjecture proposes that the antisymmetric component of the L-function generates a Hamiltonian, rather than dissipative, evolution. It has been verified in several examples, including the independent-particle-flow and zero-range-process settings, but is not established in the general abstract framework.