According to one expert,
the cause of genetic irregularities in many breeds of dog is not so much that dogs are being bred for looks or to meet other narrow criteria as that the breeds have relatively few founding members.
Option Elimination -
The questions test the usage of an idiom - "not so much X as Y."
E.g., He is not so much a leader as a follower. Here, we compare two characteristics of someone," a leader" and "a follower," in grammatically parallel form.
Not so much a strict teacher as a mentor, Professor Dutta encouraged his students to think critically and explore different perspectives. - Here, we are using "not so much X as Y" to contrast two qualities of Professor Dutta" - "a strict teacher" and " a mentor."
(A) the cause of genetic irregularities in many breeds of dog is not so much that dogs are being bred for looks or to meet other narrow criteria - Here, the structure is "dog is not so much X (relative clause introduced by "that.") as Y (relative clause introduced by "that.") Ok.
(B) the cause of genetic irregularities in many breeds of dog is not as much their being bred for looks or meeting other narrow criteria as much - Wrong idiom structure.
(C) it is not so much the cause of genetic irregularities in many breeds of dog that they are being bred for looks or meeting other narrow criteria as much - Wrong idiom structure.
(D) it is not so much the cause of genetic irregularities in many breeds of dog is their being bred for looks or meeting other narrow criteria so much - Wrong idiom structure.
(E) it is not so much the cause of genetic irregularities in many breeds of dog to be bred for looks or to meet other narrow criteria - not parallel. After "not so much," we have a phrase - the cause ... and after "as" we have a relative clause - not parallel.