Existence of fixed multipliers for odd-power binomial-sum congruences

For positive integers $n$ and $r$, define

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

Multiplier conjecture. There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that

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

and

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

The conjecture asks for universal, $n$-independent multipliers; the source notes that determining their best possible values is difficult and gives no resolution of the general claim.