How do you pronounce "archmage"?

Where arch- is a prefix indicating the top of a hierarchy, it's pronounced /-tʃ/ rather than /-k/. Cf archbishop, archdeacon and arch-criminal.

Where the Greek archi- prefix is retained in English, it has the k sound: architect, archiepiscopal, archidiaconal.

See Lexico:

arch-
Chief; principal.
Pronunciation /ɑːtʃ/

(In passing, archmage is a curious mixture of Greek archi- and Latin magus. Not a good thing.)

I checked a few online dictionaries like Wikitionary, YourDictionary, Wordnik, but none of them gives its pronunciation. I didn't find 'archmage' in other online dictionaries.

There are some guidelines for how to pronounce arch-:

• In words that were compounded in Greek, arch- is pronounced with a /k/ as in archangel, architecture, archive etc.
• Words in which the arch- is followed by a consonant, it's pronounced with /t͡ʃ/ as in archbishop, archduke etc.
• Compound words in which the arch- is added to an English word, it's usually pronounced with /t͡ʃ/ such as archenemy (from Greek arch- and English enemy).

I believe archmage belongs to the second category and is pronounced with /t͡ʃ/.