After Reading again the answer to this question and the answer to this question,
I am wondering if the language $L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in $M(x)$running $M$ doesn't move left $\}$ can be decide as well.
Maybe we can use the same method as in the last question above, and somehow locate a path which will lead us to this conclusion, by examining the transitions and etc.
What do you think?