$$X =\{\langle M, w\rangle \mid\text{$M$ has one tape and never modifies portion of the}$$ $$\text{tape that contains the input $w$}\}$$
And my proposition: Let $@$ will be character such that there is no $@$ in alphabet. Let suppose, that $X$ is decidable, so we try reduce $A_{TM}$.
Let $R$ will be machine that decides $X$, and $S$ decides $A_{TM}$
$A_{TM} = \{\langle M, w\rangle|M\ accepts\ w\} $
$S\text{ on input }= \langle M, w\rangle:$
$ \text{(1) We construct TM T:}$
$\text{ (1.1) Simulate $M$ on input $w$}$
$\text{ (1.2) When it accepts write @ on the left-most ceil of tape}$
$\text{(2) Run $R$ on input $\langle T, w\rangle$}$
$\text{(3) Accept when $R$ accepts, and reject when $R$ rejects}$
What about this solution ?