Firefox and fcitx-mozc

I've asked this on the Tokyo Linux list--although,as its name implies, most use Linux, there are several folks with FreeBSD knowledge, but none had the answer.

I use fcitx-mozc for Japanese input. I have a few variables set in my .xinitrc that work for every program save firefox. The .xinitrc contains
export XMODIFIERS=@im=fcitx
export GTK_INPUT_MODULE=fcitx
export QT_IM_MODULE=xim
export LC_CTYPE=en_US.UTF-8

I use dwm and openbox as my window managers. With Chrome, Libreoffice, urxvt, and thunderbird, this is fine, if I need to input Japanese, I hit the default ctl+space and input changes from English to Japanese. However, with Firefox, this has no effect.

I can work around it starting Firefox with LC_CTYPE=ja_JP.UTF-8 firefox. Whether starting from terminal, or having console messages saved, I see no indication of the error. (I don't know if there's a better way, though.) I can also, if I really need Japanese input in a browser, (which is so infrequent that I didn't even realize I had the problem on 10.x till I tested on a fresh 11.x install) either use the LC_CTYPE setting mentioned above, or use Chrome or Opera, so this is not a big deal, and I'm not willing to spend a lot of time or energy on it, but, if someone knows a solution that doesn't require too much effort, or (again, not requiring a lot of effort) a reasonable way to try to solve the issue, I'd be appreciative.