Scaling conjecture for filling functions of Carnot groups

Let $G$ be a Carnot group, and let $s_t$ denote the scaling by $t$ on $G$. For a map $\beta:S^k\to G$, write $\operatorname{vol}(\beta)$ for its volume and $\delta^k$ for the relevant $k$-dimensional filling function.

Scaling conjecture. There is a map $\alpha:S^k\to G$ such that

$$\delta^k(s_t(\alpha))\sim \delta^k(\operatorname{vol}(s_t(\alpha))).$$

The conjecture asserts that, in every Carnot group, some family of scaled cycles realizes the filling function at the scale prescribed by its volume. It is motivated by the preceding examples, where difficult fillings arise from scaling simple cycles rather than from topological or geometric complexity; the supplied text gives no resolution.