  We develop a full multichannel spin model in momentum space to investigate
three-body recombination of identical alkali-metal atoms colliding in a
magnetic field. The model combines the exact three-atom spin structure and
realistic pairwise atom-atom interactions. By neglecting the interaction
between two particles when the spectating particle is not in its initial spin
state we arrive at an approximate model. With this approximate model we achieve
excellent agreement with the recent precise measurement of the ground Efimov
resonance position in potassium-39 close to 33.58 G [Chapurin $et$ $al$., Phys.
Rev. Lett. 123, 233402 (2019)]. We analyze the limitations of our approximation
by comparing to the numerical results for the full system and find that it
breaks down for Feshbach resonances at larger magnetic fields in the same spin
channel. There the relevant three-body closed channel thresholds are much
closer to the open channel threshold, which enhances the corresponding
multichannel couplings. Therefore the neglected components of the interaction
should be included for those Feshbach resonances.
