This is the brutally hard Tatooine Sunset Sudoku puzzle by Philip Newman ... except the Noble Happy Star has goofed! Two of the digits have been swapped and there are multiple solutions. Fortunately, you are not required to solve the whole puzzle. All you need is six measly digits – or more precisely – their sum.
Let the sum of the six gray cells be X. Your task is to determine all possible values of X.
Acknowledgements: I wish to thank Simon Anthony and Mark Goodliffe for uploading the original Tatooine Sunset puzzle and solution. Thanks also to Tom Collyer and Sam Cappleman-Lynes for further discussion of relevant solving techniques.
EDIT: In response to Bass's comment: the original Tatooine Sunset puzzle has r3c2=8, r4c8=3 and this has a unique solution.