The forcing obstruction to embedding large abelian C*-algebras into the reduced product

Let $A$ be an abelian and nonseparable $\mathrm{C}^*$-algebra, and let $\mathbb{E}_A$ be the forcing notion associated with $A$. The density character of $A$ is greater than $2^{\aleph_0}$ when

$$\operatorname{dens}(A)>2^{\aleph_0}.$$

Forcing obstruction conjecture. The poset $\mathbb{E}_A$ forces that $A$ does not embed into $\ell_\infty/c_0$.

This is proposed as a step toward understanding when C*-algebras can be gently placed into the Calkin algebra. The supplied text gives no evidence that the claim has been resolved.