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Yes, I'd assume so that unless they say, it's regular. I own CX237's (one version older CX238) and can say that even with the regular size, it's the widest shoe I've tried, the little mesh panel on the lateral side of the shoe generates a lot of give. They're stiff as you like, though I do think this comes at the expense at sole thickness with these (though rn I guess that's last of your worries).
This page has all the info on Lake sizing:
https://ridelev.com/pages/lake-cycling-shoe-size-guideInteresting I've tried CX-332's in a 44.5 wide (my CX-237's are a 44.5 regular) and I found them to be too short, so now going to give the CX-332's a go in 45, as when it gets to summer I find the CX-237's a little bit loose with summer socks and doing harder stuff.
As Chellis says, best thing to do is try some one and ideally on a bike. I've found walking in cycling shoes and just trying them on is completely different to actually riding in them. Sounds obvious, but might need saying.
The 238’s have a regular and wide version, the page doesn’t mention so assume regular? The 332 on the other hand have a wide option in stock