Quantitative nullhomotopy conjecture via Sullivan-model depth

Let $X$ have dimension at most $n$, let $Y$ be simply connected, and let $f:X\to Y$ be nullhomotopic with Lipschitz constant at most $L$. Let

$$0=V_0\subset V_1\subset\cdots\subset V_q$$

be a filtration of the indecomposables in dimensions at most $n$ of the Sullivan minimal model of $Y$ such that $dV_i\subseteq\mathbb{Q}\langle V_{i-1}\rangle$; let $q$ be the minimal possible depth of such a filtration. Quantitative nullhomotopy conjecture. The map $f$ has a nullhomotopy of thickness $O(L)$ and width $O(L^q)$. The conjecture refines quantitative homotopy bounds by tying the width exponent to the rational homotopy type of $Y$; the source reports that the cases $q=0$ and $q=1$ are proved.