Vanishing conjecture for higher path homology of circulant digraphs

Let $\vec{C}_n^S$ be the circulant digraph with connection set

$$S=\{1,\gamma_1,\ldots,\gamma_{d-1}\},\qquad 1<\gamma_1<\cdots<\gamma_{d-1}<n/2.$$

Higher path-homology vanishing conjecture. The path homology of $\vec{C}_n^S$ vanishes in every degree at least three:

$$H_m^{\mathrm{path}}(\vec{C}_n^S)=0\qquad\text{for all }m\geq 3.$$

The authors report that their computations produce no nontrivial path homology in degrees greater than $2$. The conjecture asserts that this pattern holds for all connection sets of the stated form; its general validity remains open.