ABPS conjecture for enhanced L-parameters

Let ${\mathcal H}$ be a connected reductive group over a local non-archimedean field. Let $\mathfrak{Lev}({\mathcal H})$ represent the conjugacy classes of Levi subgroups, let $W({\mathcal H},{\mathcal L})$ be the corresponding Weyl group, and let $\Phi_{e,\zeta_{\mathcal H}}({\mathcal H})$ and $\Phi_{{\rm cusp},\zeta_{\mathcal H}}({\mathcal L})$ denote the enhanced and cuspidal enhanced L-parameter sets with the specified central character. ABPS conjecture. There exists a commutative bijective diagram

\xymatrix{ {\rm Irr} ({\mathcal H}) \ar@{<->}[r] \ar@{<->}[d] & \Phi_{e,\zeta_{\mathcal H}} ({\mathcal H}) \ar@{<->}[d] \\ \bigsqcup_{{\mathcal L} \in \mathfrak{Lev}({\mathcal H})} \big( {\rm Irr}_{\rm cusp} ({\mathcal L}) /\!/ W({\mathcal H},{\mathcal L}) \big)_\kappa \ar@{<->}[r] & \bigsqcup_{{\mathcal L} \in \mathfrak{Lev}({\mathcal H})} \big( \Phi_{{\rm cusp},\zeta_{\mathcal H}} ({\mathcal L}) /\!/ W({\mathcal H},{\mathcal L}) \big)_\kappa }

where the right-hand map is the stated bijection for enhanced L-parameters, the upper horizontal map is a local Langlands correspondence, the lower horizontal map is induced from local Langlands correspondences for cuspidal representations, and the left-hand map is the ABPS bijection. The conjecture proposes compatibility of the representation-theoretic and Galois-side extended-quotient descriptions; the paper proves the right-hand bijection but does not establish the full diagram in general.