The irrationality conjecture for the fifth zeta value and its reciprocal

The Riemann zeta function is denoted by $\zeta$, and $\zeta(5)$ is its value at $5$; $1/\zeta(5)$ denotes its reciprocal. Irrationality conjecture. The real numbers $\zeta(5)$ and $1/\zeta(5)$ are irrational. The rationality or irrationality of the fifth odd zeta value is stated in the surrounding text as unknown, so this claim remains open.