The arithmetic transfer conjecture for odd type $(r,0)$

Let $r$ be odd with $0\leq r\leq n$, and let $\widetilde{\mathcal{N}}^{[r]}_n$ be the formal scheme defined by the Cartesian construction above, with arithmetic intersection number

$$\left\langle {\widetilde{\mathcal{N}}^{[r]}_{n}},g{\widetilde{\mathcal{N}}^{[r]}_{n}}\right\rangle_{\mathcal{N}^{[r]}_{n}\times\mathcal{N}_{n+1}}.$$

Arithmetic transfer conjecture for odd type $(r,0)$. There exists $\varphi'\in C_c^\infty(G')$ transferring to $(c_r^2\\,\mathbf{1}_{K_n^{[r]}\times K_{n+1}},0)$ such that, for every matched pair $\gamma\in G'(F_0)_{\mathrm{rs}}$ and $g\in G_{W_1}(F_0)_{\mathrm{rs}}$,

$$\left\langle {\widetilde{\mathcal{N}}^{[r]}_{n}},g{\widetilde{\mathcal{N}}^{[r]}_{n}}\right\rangle_{\mathcal{N}^{[r]}_{n}\times\mathcal{N}_{n+1}}\log q=-\frac12\operatorname{\partial Orb}(\gamma,\varphi').$$

Moreover, every such transfer admits a correction function $\varphi'_{\mathrm{corr}}$ giving the corresponding identity with the additional term $-\operatorname{Orb}(\gamma,\varphi'_{\mathrm{corr}})$. This is open beyond the cases discussed in the paper; the density conjecture is stated to imply the correction-function formulation from the first part.