This is fine
NEC 400.7(A)(1) is what provides the general permission for pendants of flexible cord:
400.7 Uses Permitted.
(A) Uses. Flexible cords and cables shall be used only for the following:
(1) Pendants.
While MWBCs are not specifically stated as permitted in Article 400, the MWBC-specific neutral ampacity text in 400.5(A):
A neutral conductor that carries only the unbalanced current
from other conductors of the same circuit shall not be required
to meet the requirements of a current-carrying conductor.
In a 3-wire circuit consisting of two phase conductors and
the neutral conductor of a 4-wire, 3-phase, wye-connected system, a common conductor carries approximately the same current as the line-to-neutral currents of the other conductors and shall be considered to be a current-carrying conductor.
On a 4-wire, 3-phase, wye circuit where more than 50 percent of the load consists of nonlinear loads, there are harmonic currents present in the neutral conductor and the
neutral conductor shall be considered to be a current-carrying conductor.
is a strong clue that MWBCs can be routed over cordage using any of the provisions in NEC 400.7(A). (Also, range and dryer circuits are MWBCs by the Art. 100 definition of the term, and routinely run over cords to from wall-socket to final desination.)
So, as long as the pendant is strain relieved properly as per 400.10:
400.10 Pull at Joints and Terminals. Flexible cords and
cables shall be connected to devices and to fittings so that
tension is not transmitted to joints or terminals.
and routed into the receptacle box as per 400.14:
400.14 Protection from Damage. Flexible cords and cables
shall be protected by bushings or fittings where passing
through holes in covers, outlet boxes, or similar enclosures.
and 314.23(H)(1):
(H) Pendant Boxes. An enclosure supported by a pendant
shall comply with 314.23(H)(1) or (H)(2).
(1) Flexible Cord. A box shall be supported from a multiconductor cord or cable in an approved manner that protects the conductors against strain, such as a strain-relief connector threaded into a box with a hub.
the NEC takes no issue with this setup. (Think of it like a cord pendant dropped from a busway in an industrial plant, only with ordinary wiring methods to a L14-20 instead of the busway and busway plug.)