Furdui–Sîntămărian's consecutive harmonic-number product conjecture

Let $q\geq 1$ be an integer, and let $H_k=\sum_{j=1}^k\frac{1}{j}$ denote the $k$th harmonic number. For rational numbers $\alpha_{q,2},\alpha_{q,3},\ldots,\alpha_{q,q+2}$, consider the corresponding linear combination of zeta values. Furdui–Sîntămărian's conjecture.

$$\sum_{k=1}^\infty \frac{H_k H_{k+1}\cdots H_{k+q}}{k(k+1)\cdots(k+q)} = \alpha_{q,2}\zeta(2)+\alpha_{q,3}\zeta(3)+\cdots+\alpha_{q,q+2}\zeta(q+2).$$

The conjecture predicts that every such consecutive-product series is a rational linear combination of zeta values of weights from $2$ through $q+2$. The case $q=2$ was computed previously, while the paper's $q=3$ evaluation produces a term involving $\zeta(2)\zeta(3)$ and therefore does not presently establish or disprove the conjecture.