Directional isoperimetric conjecture for cycles in Euclidean space

Let $z$ be a closed $k$-cycle in $\mathbb{R}^n$. For an $m$-tuple $J$ of distinct coordinate indices, let $P(J)$ be the coordinate $m$-plane spanned by the corresponding axes, and let $\operatorname{Vol}_J(C)$ denote the geometric-multiplicity volume of the projection of an $m$-chain $C$ to $P(J)$. Suppose that $z$ bounds a $(k+1)$-chain $y$ with $\partial y=z$.

Directional isoperimetric conjecture. There is such a $(k+1)$-chain $y$ for which, for every $(k+1)$-tuple $I$,

$$\operatorname{Vol}_I(y) \leq C(n)\left[\prod_{J\subset I}\operatorname{Vol}_J(z)\right]^{\frac{1}{k}}.$$

This would generalize the Loomis–Whitney directional estimate and refine the Federer–Fleming isoperimetric inequality by controlling each directional volume of a filling in terms of the directional volumes of the boundary cycle. The source presents this as a natural conjecture; no resolution is supplied in the given text.