Sharing a Matomo tracking cookie across subdomains

Is there a way to share and use the same Matomo tracking cookie across subdomains?
My setup: subdomain1 is tracked and analyzed in Matomo website 1, and subdomain2 is tracked and analyzed in Matomo website 2.
Matomo cookies are set for the second-level domain:
_paq.push(["setCookieDomain", "*.example.com"]);

Unfortunately, the tracker code creates different cookies for each website:
_pk_id.1.bdda and _pk_ses.1.bdda for Matomo website 1
_pk_id.2.bdda and _pk_ses.2.bdda for Matomo website 2
The cookie names and the cookie values are different on both sites.
Is there a way to share the same cookie on all subdomains, i.e. omit the website id as part of the cookie name and use cookie names like_pk_id.bdda and _pk_ses.bdda?

The goal is to use the same visitor id in all subdomains without the need of applying cross domain linking.

Thanks in advance!