Zudilin's strengthened integrality conjecture for Apéry-type sequences

For $n\ge 0$, let ${\bf u}_n$ and ${\bf v}_n$ be the sequences defined by the Apéry-type hypergeometric series ${\bf S}_{n,4,2,1,1}(1)={\bf u}_n\zeta(4)-{\bf v}_n$. Define

$$\Phi_n=\prod_{\{n/p\}\in[2/3,1[\,,\ p\ \text{prime}} p,$$

where $\{n/p\}$ denotes the fractional part of $n/p$, and let $\textup{d}_n$ be the associated least-common-multiple denominator. Zudilin's integrality conjecture. For every integer $n\ge 0$, the numbers $\Phi_n^{-1}{\bf u}_n$ and $\Phi_n^{-1}\textup{d}_n^4{\bf v}_n$ are integers. This sharpens the denominator bounds for the sequences used in Apéry-type irrationality constructions; the surrounding text reports it as a numerical refinement of the preceding conjecture.