  Feshbach resonances of arbitrary width are typically described in terms of
two-channel models. Within these models, one usually considers a single dressed
resonance, with the option to extend the analysis by including resonant
open-channel features that can drastically change the observed threshold
effects. For the strong $^{40}\mathrm{K}$ p-wave resonance studied in Ref.
\cite{ahmed2021}, the interplay between an open-channel shape resonance and the
Feshbach resonance could explain the unexpected nonlinear variation of the
binding energy with magnetic field. However, the presented two-channel
treatment relies on the introduction of two independent fitting parameters,
whereas the typical Breit-Wigner expression would only account for one. This
results in an effective magnetic moment that acquires a nonphysical value,
which is an indication of a major shortcoming of the two-channel model
treatment. In this study, we observe how the presence of a closed-channel shape
resonance explains the physical mechanism behind the observations and
demonstrates the need of a three-channel treatment. We introduce our novel
model as \textit{resonance facilitated}, where all coupling is mediated by the
Feshbach state, while there is no direct coupling between the additional
channel and the open channel. Notably, the resonance-facilitated structure
greatly reduces the complexity of the full three-channel model. The typical
Breit-Wigner form of the two-channel Feshbach formalism is retained and the
full effect of the added channel can be captured by a single resonance dressing
factor, which describes how the free propagation in the Feshbach state is
dressed by the added channel.
