Let $NS(T), D_+(T), D_-(T)$ denote closures of a $2$-tangle $T$ as in the picture. I conjecture that if $NS(T)$ is the unknot and one of $D_+(T), D_-(T)$ is the uknot, then $T=T_0$.

We need to assume here that all tangles and links are framed (with the "blackboard" framing, i.e. the one parallel to your screen :-) since a double crossing $T$ would be a counterexample otherwise.

Is it known? Would you have a suggestion for a proof?