Jelly, 17 bytes

!P
ÇṗLÇ⁼¥ÐfÇḢḟ1ȯ0

A monadic link taking and returning a list of the numbers

Try it online!

How?

A golfed (although independently written) version of Pietu1998's solution.

Jelly,  17  18 bytes

!P
ÇṗLÇ⁼¥ÐfÇḢḟ1ȯ0F

A monadic link taking and returning a list of the numbers (sticks to the one factorial per number option)

Try it online!

How?

A golfed (although independently written) version of Pietu1998's solution.

!P - Link 1, product of factorials: list
!  - factorial (vectorises)
 P - product

ÇṗLÇ⁼¥ÐfÇḢḟ1ȯ0F - Main link: list                       e.g. [3,2,2]
Ç               - call the last link (1) as a monad           24
  L             - length                                      3
 ṗ              - Cartesian power      [[1,1,1],[1,1,2],...,[1,1,24],...,[24,24,24]]
        Ç       - call the last link (1) as a monad           24
      Ðf        - filter keep if:
     ¥          -   last two links as a dyad:
   Ç            -     call the last link (1) as a monad     [1,2,...,24!,...,24!^3]
    ⁼           -     equal?
         Ḣ      - head
          ḟ1    - filter out any ones
            ȯ0  - or with zero (for the empty list case)
              F - flatten (to cater for the fact that zero is not yet a list)