synchronization of counters in HOTP
There is no "fresh client" with HOTP. The whole counter business is based on the idea that there is a single client, who maintains his counter which is more-or-less synchronized with the server counter. The synchronization window is just a way to cope with small unsynchronization events which come from realistic situations (e.g. your 3-year-old played with your car keys and pushed the "unlock button" a few dozen times).

If you want several clients, each client has his own counter, and the server must maintain one counter per client.