I would like to understand if the following is true: Let $X$ be a continuous bounded-support random variable over $\mathbb{R}^n$ and let $Z = N(0,I)$ be a standard Gaussian independent of $X$. Say that we know the density of $X + Z$ (i.e. the convolution of densities of $X$ and $Z$) at some open set $D \subseteq \mathbb{R}^n$. Does this uniquely determine $X$? In other words, can there be two probability distributions, $X$ and $X'$, such that $X+Z$ and $X'+Z$ have the same density on $D$?
If this is not true, then I would be interested to know whether $X$ can be uniquely recovered from the values of the densities of $\{X+Z_a\}_{a \in [1,1+\epsilon]}$ at $D$, where $Z_a \sim N(0, aI)$ independently of $X$.