Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum

The numbers $R_n$ and $W_n$ are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ $\text{and}$
W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \ \sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \ 9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \ \sum_{k=0}^{p-1}(2k+1)W_k^2 &\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad$\text{if }p>3\text{.}$ \end{align*} The first two congruences were originally conjectured by Z.-W. Sun. Our proof is based on the multi-variable Zeilberger algorithm and the following observation: $$\left(\genfrac{}{}{0px}{}{2n}{n}\right)\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{m}{k}\right)\left(\genfrac{}{}{0px}{}{k}{m-n}\right)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}\left(\genfrac{}{}{0px}{}{2k}{k}\right)\left(\genfrac{}{}{0px}{}{2m-2k}{m-k}\right)),$$ where $0\leqslant k\leqslant n\leqslant m \leqslant 2n$.