Cannot set cookies when using error_page directive, why?

[pdh0710]

error_page	400 401 402 403 404 500 502 503 504	/err.html;

location = /err.html {
	root /var/www;
	add_header Set-Cookie "error_response=${status}; path=/;";
}

==========

(Please excuse my English)

Above is a part of my 'nginx.conf'. My purpose is... If error occurred,
client browser gets
'err.html' with error_response=$status cookie. The 'err.html' has JavaScript
codes that
handles the error_response cookie and display related error messages.

However, when I tried "http://test.domain.com/not_exist.html" and other urls
make error,
the client browser got 'err.html' without error_response cookie.
But when I tried "http://test.domain.com/err.html" directly, client browser
got error_response
cookie successfully.
So, I concluded Nginx does not pass cookies when using 'error_page'
directive.

Is it a Nginx bug?
Or intentionally blocked? Why?

Posted at Nginx Forum: https://forum.nginx.org/read.php?2,271704,271704#msg-271704