FWIW:
NEC (National Electrical Code) 300.22(B) does not allow an Ethernet cable to be installed in HVAC supply air fabricated duct work for the use as a raceway.
Even the exception doesn’t allow its’ use as a raceway.
300.22(B) Ducts Specifically Fabricated for Environmental Air.
Exception: Wiring methods and cabling systems, listed for use in other spaces used for environmental air (plenums), shall be permitted to be installed in ducts specifically fabricated for environmental air-handling purposes under the following conditions:
(1) The wiring methods or cabling systems shall be permitted only if necessary to connect to equipment or devices associated with the direct action upon or sensing of the contained air, and
(2) The total length of such wiring methods or cabling systems shall not exceed 1.2 m (4 ft).
NEC does allow Ethernet cables to be installed in air return plenums provided the cabling is UL Listed plenum rated. (Or, NRTL Listed meeting UL)
FWIW, If the OP does install the Ethernet cable in the supply duct work he should install an insulated grommet or bushing in the cut entry hole in the metal ductwork to protect the outer jacket, and internal insulated conductors from being cut, damaged, by the supply air movement through the duct chaffing the cable at the sharp cut entry hole of the metal duct work.
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