We report on our recent self-consistent calculations of $K^-$-nuclear quasi-bound states using a $K^-$ single-nucleon optical potential derived from 6 different chiral meson-baryon coupled-channel interaction models. They yield quite diverse $K^-$ $1s$ binding energies and rather small $K^-$ decay widths. Next, we supplement the $K^-$ single-nucleon potential by a phenomenological $K^-$ multinucleon potential introduced recently to achieve good fits to kaonic atom data [1]. We demonstrate a decisive impact of the $K^-$ multinucleon interactions on the widths of $K^-$-nuclear states. The resulting widths are considerably larger than the corresponding binding energies. The only two models accepted by the analysis of Ref. [1] even do not yield any kaonic nuclear bound state in many-body nuclear systems with mass number $A\leq 40$.