Troubleshooting: If your "Insert image" button disappears, make sure you are not adding trailing slashes to your htaccess

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.