Definition of おくめんもなく
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(adv) boldly; audaciously; impudently; unashamedly; unabashedly
「まあその、自慢できる事でもないんだけど」というか、臆面もなく吹聴できる話でもない。
"Well, that is, it's not something I can boast of," in fact it's not something I can talk unashamedly about in public.