there is no specification in the datasheet.
I really appreciate for sharing your test data. Based on comparing humidity step response data of ADR444 with ADR4540, both should perform similarly. Both ADR444 and ADR4540 integrated circuit is enclosed within plastic package. Plastic package do absorb moisture which will induce mechanical stress on the die.
Option 1: Continue to use ADR444.
Apply conformal coating (humiseal) on PCB board in order to protect against humidity variation.
NOTE: Before apply any coating please makes sure PCB board is dry and clean.
Option 2: Transition to LS8/Ceramic/hermetic package voltage reference.
LS8 packaging helps to improve stability of voltage references.
Recommended parts with similar performance as ADR444:
LTC6655BHLS8-4.096: initial accuracy = 0.025%, Vout Tempco = 2ppm/*C
LTC6655CHLS8-4.096: initial accuracy = 0.05%, Vout Tempco = 5ppm/*C
LT6654AHLS8-4.096: initial accuracy = 0.05%, Vout Tempco = 10ppm/*C