Genčev–Rucki conjecture for multiple Apéry-like series

Let $r\in\mathbb{N}$ with $r>1$, and let $\zeta_n^\star(\{2\}_r)$ denote the multiple harmonic star sum with $r$ entries all equal to $2$:

$$\zeta_n^\star(\{2\}_r)=\displaystyle\!\sum_{n\geq n_1\geq\cdots\geq n_r>0}\displaystyle\!\frac{1}{n_1^2\cdots n_r^2}.$$

Genčev–Rucki conjecture. For every $r\in\mathbb{N}$ with $r>1$,

$$\displaystyle\!\sum_{n=1}^{\infty}\displaystyle\!\frac{\binom{2n}{n}}{n4^n}\zeta_n^\star(\{2\}_r)=2(1-4^{-r})\zeta(2r+1).$$

This conjecture concerns an explicit multiple Apéry-like series involving central binomial coefficients and multiple harmonic star sums. The source attributes it to Genčev and Rucki; no resolution is stated in the supplied text.