Density-regularity characterization by computable positive-density sets

Let $R$ be a computable integral domain. Let $d^*$ denote upper additive density and $d^*_{\times}$ upper multiplicative density, and let $IADR_R$ and $IMDR_R$ denote injective additive and multiplicative density regularity, respectively.

Density-regularity characterization conjecture. There exists a computable collection $\{B_n\}_{n=1}^{\infty}$ of subsets of $R\setminus\{0\}$ such that

$$d^*(B_n)>0\quad\bigl(d^*_{\times}(B_n)>0\bigr),$$

and

$$p\in IADR_R\quad\bigl(p\in IMDR_R\bigr)\quad\Longleftrightarrow\quad\text{for every }n\in\mathbb{N},\ p\text{ has an injective root in }B_n.$$

The computability requirement asks for an algorithm deciding, from $n\in\mathbb{N}$ and $r\in R\setminus\{0\}$, whether $r\in B_n$. The assertion is presented as a further conjectural characterization of density regularity; its general validity remains open.