Nonvanishing of the order-five obstruction outside exceptional parameter values

Let $K_{4,36}$ denote the order-five variational-equation entry obtained by transporting the jet along the cycle $\gamma=\gamma_-^{-1}\gamma_+^{-1}\gamma_-\gamma_+$. Let $(\Lambda,\lambda)$ be the parameters of the Friedmann–Robertson–Walker Hamiltonian, and let $n=0,1$ refer to the exceptional cases listed in the source's equation (exc). Nonvanishing conjecture.

$$K_{4,36}\neq 0$$

for every value of $(\Lambda,\lambda)$ except for $n=0,1$ in (exc). Hence, the order-five variational equations yield the first obstruction to integrability in $H$. The claim is presented as an open problem supported by uniform numerical evidence for $k=5$ and by the preceding non-integrability theorem.