Solution

The phrase 'Many surgeons continue to advocate (i) __________ analgesic treatment' suggests that Blank (i) denotes a type of analgesic treatment.

Further down this sentence, we see that '(i) __________ analgesic treatment' is compared with 'treatment started after surgery.'

For two things to be compared, they must be unique or opposite to one another.

This means, '(i) __________ analgesic treatment' is not started after surgery. Therefore, either this treatment is started before the surgery or during the surgery.

Thus, 'Blank (i)' should be filled with a word that either means something like 'pre-surgery' or 'during-surgery'.

With this idea in mind, we evaluate the three options for Blank (i):

A. ameliorative - this means 'that which soothes pain [or another negative condition] or makes it more bearable'. This word is a synonym of 'analgesic' and doesn't carry any connotation of time (pre - or during -). Hence, we can reject this choice.
B. incipient - this means 'beginning to exist'. The phrase 'incipient analgesic treatment' would mean 'analgesic treatment that is only just starting or beginning to exist.' The opposite of 'incipient treatment' would be 'fully-developed treatment'. Therefore, this word too doesn't convey the meaning that we are looking for.
C. preemptive - this means 'that which is done ahead in time to prevent a possible bad event from happening.' So, 'preemptive analgesic treatment' would mean 'analgesic treatment that is started before the surgery or pain occurs. In this sense, it conveys the opposite of the phrase 'analgesic treatment started after the surgery.' This word fits in our intuited meaning of Blank (i).
Let is now find out the correct option for Blank (ii).

A simpler version of the question statement is:

"Many surgeons continue to advocate X despite the fact that [ABC] have failed to demonstrate (ii) __________ that X is any better than Y."

The key words in the above sentence is 'despite'. This is a contrast word. So, it indicates that the belief of the surgeons is not supported by ABC.

So, [ABC] have failed to demonstrate __________ that X is as good as the surgeons think it to be.

The possible words that can fill Blank (ii) are: clearly, definitively, beyond doubt.

With these words in mind, we'll evaluate the answer choices for Blank (ii):

D. dispassionately - this word means 'calmly or impartially'. Clearly, it doesn't fit the context here and can be rejected.
E. unequivocally - it means 'clearly, unmistakably'. This word fits in our intuited meaning of Blank (ii).
F. intractably - it means 'stubbornly, unmanageably.' To prove or demonstrate something 'Stubbornly/unmanageably' doesn't make any sense and so, this option too can be rejected.
The complete sentence is:

Many surgeons continue to advocate preemptive analgesic treatment despite the fact that clinical studies and meta-analyses have failed to demonstrate unequivocally that such treatment reduces postoperative pain more effectively than one started after surgery.

The correct answer are C and E.