Are Set-Cookie headers removed from page cache

I would like to understand how w3-total-cache handles cookies, specifically Set-Cookie headers coming with a response. Based on some preliminary tests (see below) I now believe to understand that Set-Cookie response headers are stripped and not delivered from cache. Is my conclusion correct and is this the expected behaviour for W3 Total Cache page cache?

Test setup & Tests
Environment
WordPress version: 6.4.2
PHP version: 8.1
Active plugins:

• Code Snippets
• Jetpack
• WooCommerce
• W3 Total Cache version 2.6.1

A test page under the url /page2/
An init hook to place a test cookie:

add_action('init', function() {
   setcookie('test_cookie', '1', strtotime('+1 day'));
});

Relevant config
W3 Total Cache enabled

• Page cache enabled
• One cookie-based cache group
  • cookie: no-cache

Observed behaviour
state: all caches have been purged – first request after cache purge

curl  -s -c cookies.txt -D headers.txt https://w3dk.test.local/test2/ | \
   awk 'BEGIN{dump=0}{if(dump == 1 || $0 ~ /Performance optimized.*/){dump = 1; print $0}}' && \
   { echo "-------HEADERS---------"; cat headers.txt; }


Performance optimized by W3 Total Cache. Learn more: https://www.boldgrid.com/w3-total-cache/

Page Caching using Disk: Enhanced

Page cache debug info:
Engine:             Disk: Enhanced
Cache key:          w3dk.test.local/test2/_index_slash.html
Creation Time:      1705114168.000s
Header info:
Set-Cookie:         test_cookie=1; expires=Sun, 14-Jan-2024 02:49:28 GMT; Max-Age=86400
Content-Type:       text/html; charset=UTF-8
Link:               <https://w3dk.test.local/wp-json/>; rel="https://api.w.org/"
Link:               <https://w3dk.test.local/wp-json/wp/v2/pages/15>; rel="alternate"; type="application/json"
Link:               <https://w3dk.test.local/?p=15>; rel=shortlink


Served from: w3dk.test.local @ 2024-01-13 03:49:28 by W3 Total Cache
-->
-------HEADERS---------
HTTP/1.1 200 OK
Date: Sat, 13 Jan 2024 02:49:28 GMT
Server: Apache
Link: <https://w3dk.test.local/wp-json/>; rel="https://api.w.org/", <https://w3dk.test.local/wp-json/wp/v2/pages/15>; rel="alternate"; type="application/json", <https://w3dk.test.local/?p=15>; rel=shortlink
Set-Cookie: test_cookie=1; expires=Sun, 14-Jan-2024 02:49:28 GMT; Max-Age=86400
Upgrade: h2,h2c
Connection: Upgrade
Last-Modified: Sat, 13 Jan 2024 02:49:28 GMT
ETag: "6e424e9cfeb64252651c65b7eb79fbde"
Vary: Accept-Encoding
Referrer-Policy: no-referrer-when-downgrade
Transfer-Encoding: chunked
Content-Type: text/html; charset=UTF-8

state: subsequent request – page is already cached

curl  -s -c cookies.txt -D headers.txt https://w3dk.test.local/test2/ | \
   awk 'BEGIN{dump=0}{if(dump == 1 || $0 ~ /Performance optimized.*/){dump = 1; print $0}}' && \
   { echo "-------HEADERS---------"; cat headers.txt; }

Performance optimized by W3 Total Cache. Learn more: https://www.boldgrid.com/w3-total-cache/

Page Caching using Disk: Enhanced

Page cache debug info:
Engine:             Disk: Enhanced
Cache key:          w3dk.test.local/test2/_index_slash.html
Creation Time:      1705114168.000s
Header info:
Set-Cookie:         test_cookie=1; expires=Sun, 14-Jan-2024 02:49:28 GMT; Max-Age=86400
Content-Type:       text/html; charset=UTF-8
Link:               <https://w3dk.test.local/wp-json/>; rel="https://api.w.org/"
Link:               <https://w3dk.test.local/wp-json/wp/v2/pages/15>; rel="alternate"; type="application/json"
Link:               <https://w3dk.test.local/?p=15>; rel=shortlink


Served from: w3dk.test.local @ 2024-01-13 03:49:28 by W3 Total Cache
-->
-------HEADERS---------
HTTP/1.1 200 OK
Date: Sat, 13 Jan 2024 02:49:42 GMT
Server: Apache
Upgrade: h2,h2c
Connection: Upgrade
Last-Modified: Sat, 13 Jan 2024 02:49:28 GMT
ETag: "ee0a-60ecad4525a75"
Accept-Ranges: bytes
Content-Length: 60938
Vary: Accept-Encoding
Referrer-Policy: no-referrer-when-downgrade
Content-Type: text/html; charset=UTF-8

While the Set-Cookie: test_cookie=1; … header is present in the initial request (cache miss) the same cookie is absent when the request is served from cache.