I use a tld with multiple segments usually, this works fine for accessing the sites and if they are secured using valet secure.
However when outputting via valet links it is not marked as secure.
To recreate:
valet tld test
valet secure {a site}
valet links # To show is shown as secured
valet tld test.more.segments.com
valet links # Shows unsecured
Digging into the code it looks like the issue is in Site::getCertificates() where it strips off the last segment.
It may be better to update this to just strip the tld we have in config off the end to be more robust. I've checked and when changing the tld, the certificate paths are updated so this should work even if tld's are changed after the certificate generation.
I'll submit a PR to fix