A third-order Apery-like recursion for $\zeta (5)$

In 1978, Apery has given sequences of rational approximations to $\zeta (2)$ and $\zeta (3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery’s proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $\zeta (4)$ has been discovered. The note contains a possible generalization of the above results for the number $\zeta (5)$.