No, they just need read and/or write access to ftp.bank.com, and the ability to redirect traffic from the victim's browser.
As I understood, by initiating an FTP upload and redirecting the TLS-secured request to the FTP's data port, the entire HTTP request (including cookies in the header) are "uploaded" to the FTP site. This can then be downloaded by the attacker, and now they've exfiltrated the data like the session id (and can do a session hijack attack).
A related attack can be done by placing an HTTP response on the FTP site, and getting the victim's browser to "request" that. This allows the attacker to inject javascript.
These work with certain other protocols, too, but varies by the server and browser tolerance for unrecognized stuff in a request (eg: SMTP will ignore HTTP headers, allowing an SMTP session to be embedded in a POST request as a <textarea>; IE will ignore POP3 "junk" before the beginning of an HTTP response embedded in an email body).