Cyclic-sum conjecture for finite multiple harmonic q-series

Let $t$ be a non-negative integer, let $d_0,d_1,\dots,d_t$ be non-negative integers, and let $z_n(\mathbf{s};\zeta_n)$ denote the finite multiple harmonic $q$-series associated with an index $\mathbf{s}$ and a primitive $n$-th root of unity $\zeta_n$. In both cyclic sums, interpret $d_j=d_k$ whenever $j\equiv k$ modulo $t+1$.

Cyclic-sum conjecture. For every integer $n>r$ and every primitive root of unity $\zeta_n$, the following hold. With $r=\sum_{j=0}^{t}d_j+2t$,

$$\sum_{j=0}^{t}z_n\left( \\{1\\}^{d_j},2,\\{1\\}^{d_{j+1}},2,\dots,2,\\{1\\}^{d_{j+t}} \right)= \frac{(-1)^t}{n}\binom{n+t}{r+1}(1-\zeta_n)^{r}.$$

With $r=\sum_{j=0}^{t}2d_j+3t$,

$$\sum_{j=0}^{t}z_n\left( \\{2\\}^{d_j},3,\\{2\\}^{d_{j+1}},3,\dots,3,\\{2\\}^{d_{j+t}} \right)\in (1-\zeta_n)^r\mathbb{Q}.$$

These conjectures generalize the preceding cyclic identities for indices containing alternating blocks of $1$'s and $2$'s, or $2$'s and $3$'s. The first part predicts an exact evaluation, while the second asserts rational divisibility by $(1-\zeta_n)^r$; no resolution is supplied in the source context.