Pilehrood–Pilehrood–Tauraso conjecture for cyclic sums on 1–2 indices

For $r\in\mathbb{N}$ and $s,d_0,d_1,\ldots,d_s\in\mathbb{Z}_{\geq 0}$, define

$$C_n^r(d_0,d_1,\ldots,d_s)=\sum_{j=0}^{s}z_n\left(\{r\}^{d_j},r+1,\{r\}^{d_{j+1}},r+1,\ldots,r+1,\{r\}^{d_{j+s}};\zeta_n\right),$$

where $d_j=d_i$ if $j\equiv i$ modulo $s+1$. For $s,d_0,d_1,\ldots,d_s\in\mathbb{Z}_{\geq 0}$, set

$$k=2\sum_{j=0}^{s}d_j+3s.$$

Pilehrood–Pilehrood–Tauraso's conjecture. For any positive integer $n$ with $n>k$, one has

$$C_n^2(d_0,d_1,\ldots,d_s)\in (1-\zeta_n)^k\mathbb{Q}.$$

This is one of two conjectures proposed for cyclic sums of finite multiple harmonic $q$-series at roots of unity. The paper studies related finite multiple harmonic $q$-series and proves explicit formulas for several families, while this asserted divisibility statement is presented as a conjecture here.