Partition-regularity characterization by computable finite partitions

Let $R$ be a computable integral domain, and let $PR_R$ denote partition regularity over $R\setminus\{0\}$ for polynomial equations, with $(I)PR_R$ denoting the corresponding injective notion. A computable collection of finite partitions $\{\mathcal{C}_n\}_{n=1}^{\infty}$ of $R\setminus\{0\}$ is one for which an algorithm, given $n\in\mathbb{N}$ and $r\in R\setminus\{0\}$, determines the cell of $\mathcal{C}_n$ containing $r$.

Partition-regularity characterization conjecture. There exists such a computable collection of finite partitions $\{\mathcal{C}_n\}_{n=1}^{\infty}$, with $\mathcal{C}_n=\{C_{n,i}\}_{i=1}^{n_\ell}$, such that

$$p\in (I)PR_R\quad\Longleftrightarrow\quad \text{for every }n\in\mathbb{N},\ p\text{ has an injective root in some cell of }\mathcal{C}_n.$$

Without the computability requirement, the assertion follows from countability of the polynomials over $R$. The computable version is not resolved in general; complexity results show that it cannot be replaced by a computable or $\Pi^0_1$ characterization in settings where $PR_R$ or $IPR_R$ is $\Pi^0_2$-complete.