Sentence:  [T]he historically minded modernist poet is uncertain whether there is any excuse at all for the existence of poets. He <WSD>brazens</WSD> out the dilemma by making cruel jokes at his own expense—jokes which he expects no one to see or to laugh at if seen. 
Meaning of the word marked with <WSD> tags:  "Generally followed by '''(out)''' or '''(through)''''': to carry through in a brazen manner; to act boldly despite embarrassment, risk, etc. (from 1550s.)