HTTP requests in persistent TCP connection routes traffic to the disabled server

Our Current Setup

We are using Cookie Insert method for session persistence. So LTM adds "BigipServer*" Cookie in the http response header with value as encoded IP address and port name. Subsequent requests from the client (in our case browser) will have this cookie in the request header and this helps LTM to send the request to same server. This LTM cookie's expiry is set to session, so this cookie will be cleared when we close the browser or we expire it using iRule.

Use case we are trying to achieve

We have set of servers configured as pool members serving traffic to users who are logged in. During release time, we will release the code to new set of servers and add those servers also to the LTM pool. LTM will now have servers with both old code as well as new code. We disable all servers which has the old code so that LTM routes only the requests which already has "BigipServer*" Cookie value pointing to those servers. This will not interrupt the users who are already logged in and doing some work. All new requests (new users) will be load balanced to any of the active servers which has new code. We will ask our already logged in users to logout and login back again once they are done with the current work. We have an iRule configured to expire the LTM cookie during logout, so our expectation is that users will be connected to new servers when they are logging in again.

Problem we are facing now

Even though iRule expires the LTM cookie during logout and the cookie is not present the request header of login, users are still routed to the same disabled server when they are logging in again. Ideally, LTM should have load balanced the request to any of the active servers.

Root Cause

Upon analyzing this further with network traffic, we found that, whenever the browser has a persistent TCP connection open with LTM after logout, browser uses that existing TCP connection for sending the login request. LTM routes this login request to the same disabled server which handled the previous request even though LTM cookie is not present in the request header. If we close the TCP connection manually after logout (using CurrPosts or some other tool), the browser establishes a new connection with LTM during login and LTM load balances this requests to any active server. One option for us is to send "Connection: close" in the response header during logout, but the browser may hold multiple persistent TCP connections (I have seen browser holding even three connections) and hence closing a single TCP connection will not help. Other option is to close the browser, but we don't have that choice for reasons I cannot explain here (trust me).

Question

Why LTM routes the traffic to disabled server even though LTM Cookie is not present? Doesn't it defeat the meaning of LTM Cookie? Is it a bug in LTM?

Is there a way to enforce LTM to not keep persistent connection when the request is served by a disabled server?

Is there a way to instruct browser to close all persistent connections to LTM (not just one)?

Is there a way for LTM to close all persistent TCP connections from an IP Address? What will happen to our users connecting via a proxy server?

Is there any other option available to solve our issue?