Nginx reverse proxy + URL rewrite + websockets

I saw this Question, but it doesn't seem to work for me.

Current (working) situation is:

server {
    listen 443 ssl;

    server_name updates.example.com;
    ssl_certificate     fullchain.pem;
    ssl_certificate_key privkey.pem;

    location /update {
        proxy_pass              "http://localhost:5000";

        proxy_connect_timeout   60;
        proxy_read_timeout      60;
        proxy_send_timeout      60;
        proxy_intercept_errors  off;
        proxy_http_version      1.1;
        proxy_set_header        Host               $http_host;
        proxy_set_header        X-Real-IP          $remote_addr;
        proxy_set_header        X-Forwarded-For    $proxy_add_x_forwarded_for;
        proxy_set_header        X-Forwarded-Proto  $scheme;
        proxy_set_header        X-Client-Subject   $ssl_client_s_dn;
        proxy_set_header        X-Client-Cert      $ssl_client_cert;
    }

    location / {
        proxy_pass              "http://localhost:5001";

        proxy_connect_timeout   60;
        proxy_read_timeout      60;
        proxy_send_timeout      60;
        proxy_intercept_errors  off;
        proxy_http_version      1.1;
        proxy_set_header        Host               $http_host;
        proxy_set_header        X-Real-IP          $remote_addr;
        proxy_set_header        X-Forwarded-For    $proxy_add_x_forwarded_for;
        proxy_set_header        X-Forwarded-Proto  $scheme;
        proxy_set_header        Upgrade            $http_upgrade;
        proxy_set_header        Connection         "upgrade";
    }

}

Now I wanted to add a prefix location to nginx redirect while leaving backend untouched.

Following the above Question's Answers I tried something along the lines:

    location /manage {
        proxy_pass              "http://localhost:5001/";

        proxy_connect_timeout   60;
        proxy_read_timeout      60;
        proxy_send_timeout      60;
        proxy_intercept_errors  off;
        proxy_http_version      1.1;
        proxy_set_header        Host               $http_host;
        proxy_set_header        X-Real-IP          $remote_addr;
        proxy_set_header        X-Forwarded-For    $proxy_add_x_forwarded_for;
        proxy_set_header        X-Forwarded-Proto  $scheme;
        proxy_set_header        Upgrade            $http_upgrade;
        proxy_set_header        Connection         "upgrade";
    }

or:

    location /manage {
        rewrite                 /manage/(.*) /$1  break;
        proxy_pass              "http://localhost:5001";
        proxy_redirect          off;

        proxy_connect_timeout   60;
        proxy_read_timeout      60;
        proxy_send_timeout      60;
        proxy_intercept_errors  off;
        proxy_http_version      1.1;
        proxy_set_header        Host               $http_host;
        proxy_set_header        X-Real-IP          $remote_addr;
        proxy_set_header        X-Forwarded-For    $proxy_add_x_forwarded_for;
        proxy_set_header        X-Forwarded-Proto  $scheme;
        proxy_set_header        Upgrade            $http_upgrade;
        proxy_set_header        Connection         "upgrade";
    }

Actually both those "kind of" work, as the correct page is displayed, but both fail to update it via websocket. I see no error in application, but whatever I do on page does not trigger anything in backend application (a python RemI App). Apparently websocket connection is not carried over correctly.

What am I missing?