The tensor-product embedding conjecture for algebraic central division algebras

Let $K$ be a field such that ${\rm abrd}_{p}(K)<\infty$ for every $p\in\mathbb P$, and let $R$ be an algebraic central division $K$-algebra. A $K$-subalgebra $\widetilde R$ of $R$ is required to satisfy the following properties:

Tensor-product embedding conjecture. The algebra $\widetilde R$ is $K$-isomorphic to

$$\widetilde R\cong\bigotimes_{p\in\mathbb P}R_p,$$

where the tensor product is over $K$ and each $R_p\in d(K)$ is a $K$-subalgebra of $R$ of $p$-primary degree $p^{k(p)}$; every $K$-subalgebra $\Delta$ of $R$ that is locally finite-dimensional of at most countable dimension embeds in $\widetilde R$ as a $K$-subalgebra. In addition,

$$C_R(\widetilde R)=K,$$

where $C_R(\widetilde R)=\{c\in R: c\widetilde r=\widetilde r c,\ \widetilde r\in\widetilde R\}$, and for each $p\in\mathbb P$, $k(p)$ is the maximal integer for which there exists $\rho_p\in R$ such that $p^{k(p)}$ divides $[K(\rho_p):K]$.

This conjecture concerns the structure of algebraic central division algebras over fields of finite absolute Brauer $p$-dimensions. It would provide a single tensor-product subalgebra containing every locally finite-dimensional countable-dimensional subalgebra, together with a precise description of its $p$-primary factors and centralizer; the supplied material gives no resolution status.