Ben-Zvi's singular-support conjecture for chains

Let $X$ be a smooth ambient space and let $Z(f)\subset X$ be the zero locus of $f$. Model chains on $Z(f)$ using currents on $X$, and call a current smooth when it is represented by a smooth chain on $X$. Ben-Zvi's conjecture. There is a good notion of singular support for chains on $Z(f)$ which measures the failure of a chain to be represented by smooth chains on $X$. Such a notion would extend the role of singular support from currents to chains on singular spaces; the source presents this as a conjectural direction and gives no resolution.