Will your character reply to a post from Kaz about it? That's a maybe. If he did, it wouldn't be with enthusiasm.

Will your character willingly carry around thread for a few days?
No. He doesn't want to be tracked and doesn't really trust Lark now.

Will your character notice Kaz using sleight of hand to plant a spool of thread on them, and will they then notice him stealing it back after 3 days?
I don't think so! He's fairly easy to manipulate.