The notoriety only a hypochondriac would enjoy (3,4)

I believe the answer is:

ill fame

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'the notoriety' is the definition.
(synonyms)

I don't understand the remainder of the clue.

(Other definitions for ill fame that I've seen before include "bad report" , "Disrepute" , "notoriety" , "celebrity of sorts" , "A house of . . . . . . . is a brothel" .)

I've seen this clue in The Globe and Mail.

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