Not possible: Doing so will block the (large) monitor. The room is indeed not ideal for sound; I'm trying to make lemonade with the lemons I've got.
Strictly outside won't work - it'd be in a doorway on one side, and another area I'd need to be able to walk through regularly on the other. I can put them at the end of the alcove (where the
other end of the table is), and doing that is absolutely worth trying.
Carpet that's 5 cm thick?!? My brain is exploding thinking about what that'd feel like to walk on. Probably broken ankles...
At least a few decades ago, "a carpet that covers the entire floor" was referred to as "wall to wall carpet." I actually haven't heard "wall to wall carpet" as a phrase for, well, decades, so your phrase works perfectly. It is, however, a perfect chance to repost this comic:
View attachment 494977
And that's considering you now have another 8-10 batts of ROCKWOOL (← Autocorrect did that bit of branding?!?) to make even more panels.
The room (as a whole) is roughly 10 feet / 3m on a side. Other practicalities for the layout are why the desk & monitor are in the alcove, instead of, well,
anywhere else.
To me, headphones are kind of a huge hassle as well. For one, I wear headphones 8 hours a day for half of my workdays, and by the time my workday is done, I can't wait to be rid of them.
Being able to use speakers and take the #%@& headphones off has been the best part of the home office. My most comfortable open-back headphones (Sennheiser HD-6XX's), while a world better than anything else I have, they're still fairly stuffy, hot, and uncomfortable to me. I also have issues with IEM's and other headphone form factors.
Another factor is I was born with tinnitus (I've never known a day without hearing it), and most headphones use either passive isolation or active noise cancelling... which makes my tinnitus
worse.
So yeah... headphones are on paper a good idea, but far less so in my
particular case.
There is carpet covering the entire floor, but that's a good idea: I am
definitely going to try putting up pillows, plush animals, piles of just-cleaned laundry, etc. to try out how that works. I realize they don't have the
mass of some acoustic panels, but it'd definitely give a hint as to what may or may not work.
I mean, yeah, there may be reflections, but as I said originally, perhaps not with enough emphasis: I can play exclusively through the center, and it sounds fine. Then I can disconnect the center and use the stereo L/R speakers, and that's when I hear the echoing. I'm not sure what disconnecting the center speaker would help as I've already tested that option. I'm game to learn, of course, but I just don't see it with what I've tested & observed.