There are some niceties here, but I think this is a little thin on the security aspects of the scheme: it's not clear how users establish the authenticity of transitively received petnames, for example. More fundamentally, there's a factor outside of Zooko's triangle: trust isn't really transitive[1]. I trust my doctor and my doctor trusts their sibling, but I don't necessarily trust their sibling. With that being sa…
There's an associated paper that goes through implementing a petname system in a simple chat application. Petnames compose well with object capability security. https://files.spritely.institute/papers/implementation-of-pe...
There are ways to (dis)intermediate that trust (like a PKI), but the shape of that PKI or other technique is itself a question of decentralization, security, etc. I think that's a very hard underlying problem that the petname design needs to at least offer some opinions on in order to make claims about security.