Continuous Hirsch conjecture

For a polytope $P$ of dimension $d$ defined by $n$ inequalities and a linear objective function $c$, let $\lambda_c(P)$ be the total curvature of its central path. Let $\Lambda(n,d)$ be the largest such total curvature over all such $P$ and $c$. Continuous Hirsch conjecture. There is a constant $K$ such that

$$\Lambda(n,d)\leq Kn$$

for all $n$ and $d$, equivalently $\Lambda(n,d)\in O(n)$. The previously proposed constant and linear-in-dimension bounds for central-path curvature were disproved by constructions showing exponential growth in $d$ and, for fixed $d\geq 2$, $\liminf_{n\to\infty}\Lambda(n,d)/n\geq\pi$.