De Klerk–Dobre conjecture on the subtour LP for circulant TSP

Let $c_1,\ldots,c_d$ be the edge costs of a circulant instance and let $\phi$ be an associated stripe permutation. Let $\operatorname{OPT}_{\mathrm{LP}}$ denote the optimal value of the subtour LP, and let $\operatorname{VDV}$ denote the combinatorial lower bound

$$\operatorname{VDV}=\left(\sum_{i=1}^{\ell}(g_{i-1}^{\phi}-g_i^{\phi})c_{\phi(i)}\right)+c_{\phi(\ell)}.$$

De Klerk–Dobre conjecture. For every circulant TSP instance,

$$\operatorname{VDV}=\operatorname{OPT}_{\mathrm{LP}}.$$

De Klerk and Dobre supported this equality with numerical experiments; the subtour LP is known to be at least as strong as the VDV bound, while equality in general remains open.