Some museums regard themselves as keepers rather than owners of art, responsible for conserving it in the present and letting it go where circumstances are auspicious to do it in the future.

A. letting it go "where" circumstances are auspicious to do "it" in the future - Incorrect - It has no antecedent. Also, where is incorrectly used here. Where refers to place.
B. letting "them" go "where" circumstances are auspicious to do so in the future - Incorrect - Them is wrong here. It should be used.
C. letting "them" go when circumstances are auspicious to do "it" in the future - Incorrect - Them is wrong here. It should be used, also it is incorrect here.
D. letting it go when circumstances are auspicious to do so in the future - Correct
E. letting it go when circumstances are auspicious to do 'it' in the future - Incorrect - It is wrong here.

