Simplicial-chain realization conjecture with prescribed general position

Let $C$ be a $c$-chain in $\mathbb{R}^d$ with $c<d$, and let $U$ be a finite set of points in $\mathbb{R}^d$ in strong general position with respect to $C$. Simplicial-chain realization conjecture. The union of $C$ is the support of a simplicial $c$-chain $C'$ in $\mathbb{R}^d$ such that $U$ is in general position with respect to $C'$. This concerns replacing a possibly nonsimplicial chain by a simplicial chain with the same support while preserving the required general-position condition; the supplied material gives no evidence that it has been resolved.