A series involving a product of four consecutive harmonic numbers

In correspondence with Goldbach, Euler began investigating series of the form ${\sum }_{k\ge 1}{k}^{-m}(1+2^{-n}+\cdots +k^{-n})$, which are known today as Euler sums. For the case where $n=1$ and $m\ge 2$, Euler was able to obtain a closed form in terms of zeta values. We use elementary techniques in the spirit of Euler to evaluate the series ${\sum }_{k\ge 1}\frac{H_k{H}_{k+1}{H}_{k+2}{H}_{k+3}}{k(k+1)(k+2)(k+3)},$ where $H_k:=1+\frac{1}{2}+\cdots +\frac{1}{k}$ is the $k$th harmonic number, in terms of zeta values. The closed form is a potential counterexample to a conjecture of Furdui and Sîntămărian. We relate this problem to conjectures regarding irrationality and $\mathbb{Q}$-linear independence of zeta values.