Extended non-integrality conjecture for generalized binomial sums

Let $n,r\geq 1$, let

$$\mathcal{L}=\{-3,-2,-1,2,3,4,5\},$$

and let $\mathcal{S}_{(r,n)}(\ell)$ denote the generalized binomial sum defined in the paper. Extended non-integrality conjecture. If $\ell\in\mathcal{L}$, then

$$\mathcal{S}_{(r,n)}(\ell)\notin\mathbb{Z}.$$

The conjecture extends the known non-integrality result beyond the range $r\geq n$ or $n>r\geq n/5$; it would in particular imply the existing conjecture for $\ell=2$.