Crofton-type conjecture for random Hamiltonian diffeomorphisms

Let $(M,\omega)$ be a closed symplectic manifold and let $\mathcal{D}$ be a centered, time-symmetric, exhaustive or periodically exhaustive law-defining datum. Let $L\subset M$ be a closed Lagrangian submanifold. The datum specifies an almost complex structure $J$ and the associated Riemannian metric $g$.

Crofton-type conjecture. There exist a constant $C>0$ and a smooth function $\rho:M\to\mathbb{R}_{>0}$, depending only on $\mathcal{D}$ and $L$ and satisfying

$$\int_M \rho\,\omega^n=\operatorname{vol}(M),$$

such that, for every closed Lagrangian submanifold $K\subset M$,

$$\int_{\operatorname{Ham}(M,\omega)} \#(L\cap\varphi(K))\,d\mu_{\operatorname{Ham}}^{\mathcal{D}}(\varphi)=C\cdot\operatorname{vol}_{\rho\cdot g}(K).$$

In particular, there exist constants $C',C”>0$ such that

$$C'\cdot\operatorname{vol}_g(K)\leq\int_{\operatorname{Ham}(M,\omega)}\#(L\cap\varphi(K))\,d\mu_{\operatorname{Ham}}^{\mathcal{D}}(\varphi)\leq C”\cdot\operatorname{vol}_g(K).$$

The conjecture seeks a Crofton formula for expected Lagrangian intersection counts, with a density-rescaled metric accounting for the nonuniform behavior seen in simulations. The supplied text gives numerical evidence and explains why the standard metric does not yield an exact formula, but states no resolution; the conjecture remains open.