Reporting that one of
its many challenges had been the highly publicized break, over creative differences, with the director of the film, the studio confirmed that it would suspend production on its adaptation of Joseph Conrad’s Nostromo.
Since studio is the main subject here, the use of 'their' is wrong as 'it' is used for studio. So Option D and E are eliminated on this basis.
(A) its many challenges had been the highly publicized
This option looks grammatically correct as the correct pronoun and tense is used. Past perfect tense is the correct tense to be used here as it describes an action that happened in the past and finished in the past based on which the studio would suspend the production.
Let's keep this.(B) its many challenges has been the highly publicized
This option is wrong as it uses present perfect tense which is is wrong.
INCORRECT(C) its many challenges is the highly publicized
This option is wrong as it uses present tense which is is wrong.
INCORRECT(D) their many challenges is the highly publicized
As discussed above, this is wrong.
INCORRECT(E) their many challenges had been the highly publicized
As discussed above, this is wrong.
INCORRECTOption A is the right answer.