The length-complexity conjecture for dynamical bordisms

Let $(M,X)$ be a dynamical bordism with reaching function $\rho_{(M,X)}:\mathbb{N}\dashrightarrow\mathbb{N}$, and let $g_0$ be the distinguished metric on a clean bordism, flat within each standard tube. For a flow $\varphi_t$ of $X$, with $t_n$ the time at which the trajectory from $n$ reaches the outgoing boundary, define

$$\operatorname{LenC}_{(M,g_0)}(n)=\int_0^{t_n}\lVert X_{\varphi_t(n)}\rVert_{g_0}\,dt.$$

Length-complexity conjecture. Let $f:\mathbb{N}\dashrightarrow\mathbb{N}$ be a partial recursive function. Given a Turing machine $\mathcal{G}$, for any $n\in\mathbb{N}$ let $T(n)$ be the number of steps that $M$ performs when started with input $n$ before halting. If $M_{\mathcal{G}}$ is the dynamical bordism resulting from thickening the graph $\mathcal{G}$, then

$$\lim_{n\to\infty}\frac{\operatorname{LenC}_{(M_{\mathcal{G}},g_0)}}{T(n)}=1.$$

The preceding prose motivates this claim by saying that the distinguished metric computes the Euclidean length of the flow and conjecturing asymptotic comparability with ordinary time complexity. The statement is presented in a section of refined open problems.