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
\(\text{MiNNLO}_{\rm PS}\) framework. We discuss both the computational challenges involved in parton-shower-matched simulations of \( e^+ \nu_e b \mu^- \bar{\nu}_\mu \bar{b} \) 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.

