Hey Robbie,
No, it will not be up because there is no 'gold' between r3 and r4 in your example.
The difference between -any and -all is actually shown in the cli help:
erdems@P-12# set admin-group ?
Possible completions:
+ apply-groups Groups from which to inherit configuration data
+ apply-groups-except Don't inherit configuration data from these groups
+ exclude Groups, all of which must be absent
+ include-all Groups, all of which must be present
+ include-any Groups, one or more of which must be present
Still following your example, the LSP would be UP if you configure admin-group include-any [ gold silver ], which would mean "include either gold or silver interfaces".
Cheers,