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

Let $0\leq r\leq n+1$, let $\varepsilon$ be the parity of $r$, and let $W_\varepsilon$ and $W_{\varepsilon+1}$ be the hermitian spaces of dimension $n+1$ with opposite invariants. Let $\varphi_r^{[\varepsilon]}$ be the specified Hecke function, and let $\widetilde{\mathcal{M}}^{[r]}_n$ be the corresponding formal subscheme. Arithmetic transfer conjecture for type $(0,r)$. There exists $\varphi'\in C_c^\infty(G')$ transferring to $((\mathbf{1}_{K_n^{[0]}}\otimes\varphi_r^{[\varepsilon]},0))$ such that, whenever matched regular semisimple elements $\gamma\in G'(F_0)$ and $g\in G_{\varepsilon+1}(F_0)$ are given,

$$\left\langle \widetilde{\mathcal{M}}^{[r]}_n,g\widetilde{\mathcal{M}}^{[r]}_n\right\rangle_{\mathcal{N}^{[0]}_n\times\mathcal{N}^{[r]}_{n+1}}\log q=-\frac12\operatorname{\partial Orb}(\gamma,\varphi').$$

The conjecture supplies the missing test function in the type $(0,r)$ arithmetic transfer problem; explicit candidates are available in some even cases, while the general assertion remains open.