Small-index multipliers for binomial-sum congruences

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

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

Small-index multiplier conjecture. Set

$$a_3=3,\ a_5=15,\ a_7=21,\ a_9=15,\ a_{11}=33,\ a_{13}=1365,\ a_{15}=3,$$

and

$$b_2=12,\ b_3=4,\ b_4=60,\ b_5=20,\ b_6=84,\ b_7=28,\ b_8=60,\ b_9=20,\ b_{10}=132,\ b_{11}=44,\ b_{12}=5460.$$

Then the congruences

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

and

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

hold for the corresponding listed indices. This is a numerical proposal for particular multipliers, following the general multiplier conjecture; the supplied text does not establish these congruences.