Our SEO people told us that all paths should have trailing slashes, so this was added to our htaccess:

RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_METHOD} !=post [NC]
RewriteCond %{REQUEST_URI} !(.*)/$
RewriteRule ^(.*(?:^|/)[^/\.]+)$ $1/ [L,R=301]

When the above code is active, IMCE works but for one detail: the Insert Image button disappears entirely. I thought I'd document it here to make sure others don't spend hours trying to figure out the cause of this bug.