CNAME's with Cloudflare Tunnels

Hello,
I have 2 domains that needs to point to the same webpage.
The first one is hosted/proxied within Cloudflare.
So I have setup a tunnel for the subdomain ping

example.com to my server.
But now I have another domain with name severs that I cannot change.
They created a CNAME record to ping.example.com.
SO ping.example-2.com is configured to ping.example.com.
But Cloudflare only gives an 404 error/blank page back on ping.example-2.com (not on ping.example.com, there the webpage is shown).
Is there a possibility to make it such that this works?

You might be able to do with locally configured tunnels, as I am not sure how much validation they do of if a hostname is in your account. I doubt it would work with remote tunnels.