Refined leading-term conjecture for regular supercuspidal connections

Let $G_1,\ldots,G_p$ be twisted Levi subgroups, let $\phi_{(i)}:\fa_i\to\bC$ be refined leading terms, let $X_{(i)}\in\check{\fa}_i$ be their duals, and let $A'$ and the central supports be defined from the associated subgroup $J$ and induced modules. Refined leading-term conjecture. (i) The quotient central support satisfies

$$\operatorname{Spec}\fZ_{\fj^+}/A'\simeq p^{-1}(\operatorname{Loc}(X_{(i)},R,\theta)).$$

The associated Levi subgroups are dual to $G_1,\ldots,G_p$. (ii) For a character $\tphi$ of $J$, $\operatorname{Spec}(\fZ_{\fj,\tphi})=p^{-1}(\nabla)$ for a unique formal connection with the stated slopes, dual Levi subgroups, and dual refined leading terms. (iii) These local systems exhaust the irreducible formal connections whose wild-inertia image has maximal-torus centralizer. (iv) $G(F)$-conjugate pairs $(S,\phi)$ and $(S',\phi')$ yield induced modules with the same central support. This is presented as a refinement for a large family of regular supercuspidal representations and as a possible approach to proving the broader correspondence; its resolution is not stated.