Let $NS(T), D_+(T), D_-(T)$ denote closures of a $2$-tangle $T$ as in the picture. $T_0$ (below) is called trivial. 

**Question**: Suppose that $NS(T)$ and $D_+(T)$ are trivial knots.
Does it imply that $T$ is either $T_0$ or $T_2=$ the 2-braid with 2 negative crossings?


[![enter image description here][1]][1]




  [1]: https://i.sstatic.net/t7rKwm.jpg