Shen et al.'s homotopy conjecture for total cut complexes of powers of cycles

Let $C_n^r$ denote the $r$th power of the cycle on $n$ vertices, and let $\Delta_d^t(C_n^r)$ denote its total $d$-cut complex. For $d\geq3$ and $n\geq(2d-1)r+1$, Shen et al.'s conjecture.

$$\Delta_d^t(C_n^r)\simeq\mathbb{S}^{n-2d}.$$

The paper states that this conjecture is partially resolved: the asserted homotopy equivalence is proved for $r\geq2$ and $n\geq2rd$ for all $d\geq2$, while the full stated range remains unresolved.