The Lie algebra arithmetic transfer conjecture

Let $\mathfrak{s}$ be the Lie algebra symmetric space and let $\mathfrak{u}_0$, $\mathfrak{u}_1$ be the matching unitary Lie algebras. Let $\mathfrak{k}_0$ be the stabilizer of the specified nearly $\pi$-modular lattice, and for $x\in\mathfrak{u}_1(F_0)$ define

$$\ell\text{-}\operatorname{Int}(x):=\operatorname{length}(\Delta\cap\Delta_x).$$

Lie algebra arithmetic transfer conjecture. (a) There exists $\phi'\in C_c^\infty(\mathfrak{s})$ transferring to $(\mathbf{1}_{\mathfrak{k}_0},0)$ such that, whenever matching $y\in\mathfrak{s}(F_0)_{\mathrm{rs}}$ and $x\in\mathfrak{u}_1(F_0)_{\mathrm{rs}}$ have artinian $\Delta\cap\Delta_x$,

$$2\omega(y)\operatorname{\partial Orb}(y,\phi')=-\ell\text{-}\operatorname{Int}(x)\cdot\log q.$$

(b) For every such $\phi'$ there exists $\phi'_{\mathrm{corr}}\in C_c^\infty(\mathfrak{s})$ such that

$$2\omega(y)\operatorname{\partial Orb}(y,\phi')=-\ell\text{-}\operatorname{Int}(x)\cdot\log q+\omega(y)\operatorname{Orb}(y,\phi'_{\mathrm{corr}}).$$

This is the Lie-algebra analogue of arithmetic transfer, with the artinian-intersection condition ensuring that the length is finite.