Continuous $d$-step Conjecture for central-path curvature

Let $\Lambda(n,d)$ denote the largest total curvature of a central path over polytopes defined by $n$ inequalities in dimension $d$ and over all linear objectives.

Continuous $d$-step Conjecture. The function $\Lambda(2d,d)$ grows linearly in its input; that is,

$$\Lambda(2d,d) \in O(d).$$

The source states that this conjecture is equivalent to the Continuous Hirsch Conjecture. The supplied material gives no resolution of either conjecture.