Problems using Korean from the second virtual desktop when using desktops.exe (IME?)

Question

I don't think it was like this in the past, but from some point on, when I use sysinternals desktops.exe, I can't use Korean from the second virtual desktop. Even if I press the Korean-English switch key, only English is available, and the IME icon does not appear at all in the task bar. I've tried several things on my own, but I haven't been able to find a solution yet. In this case, is there any action or solution I should do?

The usage environment is applicable to both Win10 and Win11.