Let $\mathcal{R}_q = \mathbb{Z}_q/\langle x^n + 1 \rangle$, with $n$ a power of $2$. Suppose that we sample $\mathbf{r} \leftarrow \mathcal{R}_q^m$ uniformly at random with the property that $0 < ||\mathbf{r}|| \leq \beta$. Given $(\mathbf{A} \leftarrow \mathcal{R}_q^{n \times m}, \mathbf{s} = \mathbf{Ar})$, find $\mathbf{r}'$ such that $\mathbf{Ar'} = \mathbf{s}$ and $0 < ||\mathbf{r}'|| \leq \beta$. This is problem known as Search Module-LWE problem.
What I want to do is proving in zero knowledge that I have such an $\mathbf{r}'$ without revealing any information about it (assuming that the verifier has access to both $\mathbf{A}$ and $\mathbf{s}$). Is it possible to do that?
