No. In the atomic swap concept, the contract ensures the funds are only "moved" when all parties have fulfilled their obligations. That's why it's called an "atomic" (undivisable) swap. So there is no need for a "box" with an additional multisig-secured "security deposit".

If one party does not carry out the accorded transaction, then the other party can withdraw the funds from the contract without suffering a loss. The only loss could be transaction fees in the original concept, but a similar contract can also be achieved off-chain (with similar mechanisms than the Lightning Network offers).