Notice that the sentence says, "nevertheless, the judges remained unmoved." So, we can tell that the inmate must have done something that might be reasonably expected to move the judges to set him free.

We can eliminate "chided" (scolded) and "snubbed" (ignore disdainfully) since neither of those negative actions would be expected to move the judges to set the inmate free.

"Conceded" (admitted that something is true) might somehow have moved judges to free the inmate but has no synonym in the list wouldn't be "humiliating."

It doesn't make sense at all that the inmate would have "drummed" (played a drum or made a rhythmic noise) the judges.

What does make sense is that the inmate "beseeched" (begged) or "supplicated" (begged) the judges but didn't move them.

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