The correct option is A that which they would
Here, the comparison is between "which backdrop they would prefer not to have" and 'which backdrop they would prefer to have'. The underlined word 'would' does not clarify what is being compared. So, option D is incorrect. Since we are comparing the two phrases, they both need to follow the same structure. In option A, 'that which they would' is a modification of the phrase 'the backdrop which they would' where 'that' demonstrates the backdrop and 'prefer to have' is not required to be repeated again. Hence, option A is the correct answer. Option B is an incomplete phrase and does not have the same structure as the first phrase. Hence, it's an incorrect option. Option C is incorrect for similar reasons, since it omits 'which' which is an important determiner that refers to the noun 'backdrop'.