One signature is not a contract

Your analysis is logical, but based on the wrong premise that a lease with one signature is somehow binding.  No, although there is such a thing called unilateral contract （mosly public reward for some desired action), a lease is never unilateral and requires two signatures.  It is called an agreement, which needs two parties to agree.

Our LZ's problem is that he doesn't know if the other party signs the lease, and it is to her advantage to sign it given LZ's intention.  

You suggestion is otherwise sound.