The maximum-degree conjecture for extreme points of the Held–Karp relaxation

Consider extreme points of the Held–Karp relaxation on $n$ vertices, and let the maximum degree of an extreme point mean the maximum degree of its support graph. The maximum-degree conjecture. The maximum degree of extreme points on $n$ vertices is exactly

$$\left\lceil\frac{n}{2}\right\rceil.$$

The paper derives examples with large support degree and presents this claim as a conjecture based on enumeration through 12 vertices; whether the upper bound holds in general remains open.