Mneimneh-type binomial-sum conjecture for multiple harmonic-star sums

Let $r\in\mathbb{N}_0$ and $n\in\mathbb{N}$. For any reals $x,y$, let ${\bf p}_{r+1}:=(p_1,p_2,\ldots,p_{r+1})\in\mathbb{N}^{r+1}$ and ${\bf m}_r:=(m_1,\ldots,m_r)\in\mathbb{N}_0^r$. Define $|{\bf p}|_j:=p_1+\cdots+p_j$ and $|{\bf m}|_j:=m_1+\cdots+m_j$, with $|{\bf m}|_0:=0$, and let $\zeta_k^\star(s_1,\ldots,s_d)$ denote the finite multiple harmonic-star sum. Mneimneh-type binomial-sum conjecture. One has

$$\begin{aligned} &\displaystyle\!\sum_{k=0}^n x^ky^{n-k}\binom{n}{k}\zeta_k^\star\left(\{1\}_{p_1-1},m_1+2,\ldots,\{1\}_{p_r-1},m_r+2,\{1\}_{p_{r+1}-1}\right)\\ &=(x+y)^n\displaystyle\!\sum_{n\geq n_1\geq\cdots\geq n_{|{\bf p}|_{r+1}+|{\bf m}|_r+r-1}\geq1} \displaystyle\!\frac{\left(\displaystyle\!\frac{y}{x+y}\right)^{\displaystyle\!\sum_{j=1}^r\left(n_{|{\bf p}|_j+|{\bf m}|_{j-1}+j-1}-n_{|{\bf p}|_j+|{\bf m}|_j+j}\right)}}{n_1\cdots n_{|{\bf p}|_{r+1}+|{\bf m}|_r+r-1}}\\ &\qquad\times\left(1-\left(\displaystyle\!\frac{y}{x+y}\right)^{n_{|{\bf p}|_{r+1}+|{\bf m}|_r+r-1}}\right). \end{aligned}$$

This conjecture would extend the paper's Mneimneh-type identities from the proved cases to the stated general parameters; its resolution is not established in the supplied text.