Guo and Liu's existence conjecture for divisibility constants

Let $n$ and $r$ be positive integers, and define

$$S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$$

Guo and Liu's existence conjecture. There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that

$$a_{2r-1}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv0\pmod{n^2},$$ $$b_r\sum_{k=0}^{n-1}kS_k^{(r)}\equiv0\pmod{n^2}.$$

The paper proves that such integers exist, thereby resolving this conjecture; it also treats explicit values of these constants in the following conjecture.