The three-torus fiber-length bound

Let $g$ be a Riemannian metric on the three-dimensional torus $T^3$, and let $f:T^3\rightarrow\mathbb{R}^2$. Fiber-length conjecture. There is a function $f$ such that, for every $y\in\mathbb{R}^2$,

$$\operatorname{Length}\bigl[f^{-1}(y)\bigr]\leq C\operatorname{Vol}(T^3,g)^{1/3}.$$

The source presents this as an open problem at the edge of its discussion of metric geometry in dimension three and as a naive conjecture related to the earlier level-set bounds.