Sun's integrality conjecture for products of consecutive generalized polynomials

Let ${\mathbb Z}^{+}$ denote the positive integers, and for $n\in{\mathbb Z}^{+}$ define

$$w_n^{(\alpha)}(x)=\sum_{k=1}^{n}w(n,k)^{\alpha}x^{k-1},$$

where

$$w(n,k)=\frac{1}{k}\binom{n-1}{k-1}\binom{n+k}{k-1}.$$

Write $w_n(x)=w_n^{(1)}(x)$, let $(a,b)$ denote the greatest common divisor of $a$ and $b$, and let $\mathbb Z[x]$ denote the ring of polynomials in $x$ with integer coefficients. Sun's product-integrality conjecture. For any $\alpha,m,n\in{\mathbb Z}^{+}$,

$$\frac{2(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}k(k+1)(k+2)\bigl(w_k^{(\alpha)}(x)w_{k+1}^{(\alpha)}(x)\bigr)^m\in\mathbb Z[x].$$

For any $m,n\in{\mathbb Z}^{+}$,

$$\frac{2(2,n)}{n(n+1)(n+2)(2x+1)^m}\sum_{k=1}^{n}k(k+1)(k+2)\bigl(w_k(x)w_{k+1}(x)\bigr)^m\in\mathbb Z[x].$$

If $n\in{\mathbb Z}^{+}$ is even, then

$$\frac{4}{n(n+1)(n+2)(2x+1)^3}\sum_{k=1}^{n}k(k+1)(k+2)w_k(x)w_{k+1}(x)\in\mathbb Z[x].$$

These conjectures extend the paper's study of integrality and divisibility phenomena for Sun's generalized polynomials; their resolution is not specified in the supplied text.