Progress on NNLO+PS predictions for top-quark pair production and decay

In this proceeding we report on recent progress toward achieving NNLO+PS predictions for off-shell top-quark production and decay in the fully-leptonic channel within the MiNNLO_PS framework. We discuss both the computational challenges involved in parton-shower-matched simulations of e⁺ν_ebμ⁻ν̄_μb̄ production and the conceptual obstacles posed by the lack of the two-loop off-shell amplitude. We present one-loop studies aimed at assessing the validity of possible approximations for the virtual contributions of the dominant double-resonant contribution.