I'm not sure whether I'd say they were antonyms but they are different- I always think of implying as something the speaker/writer does and inferring as something the reader/listener does
Just found this - if it helps - or perhaps confuses
Main Entry: hint
Part of Speech: verb
Definition: suggest; indicate
Synonyms: acquaint, adumbrate, advise, allude to, angle, apprise, bring up, broach, coax, connote, cue, drop, expose, fish*, foreshadow, give an inkling, impart, imply, infer, inform,
infer [ɪnˈfɜː]
vb -fers, -ferring, -ferred (when tr, may take a clause as object)
1. to conclude (a state of affairs, supposition, etc.) by reasoning from evidence; deduce
2. (tr) to have or lead to as a necessary or logical consequence; indicate
Perhaps I should clarify before I get my ankles bit:
I'm not saying you are wrong for I know you are right but I am saying that 'infer' is the answer and therefore right, not wrong.