Guo and Liu's congruences for the sums S_n^(r) and T_n^(r)

Let $n$ and $r$ be positive integers, let $p$ be a prime, and define

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

Guo and Liu's congruence conjecture. One has

$$\sum_{k=0}^{n-1}S_k^{(2r)}\equiv0\pmod{n^2},$$ $$\sum_{k=0}^{n-1}T_k^{(2r)}\equiv0\pmod{n^2},$$ $$\sum_{k=0}^{p-1}T_k^{(2)}\equiv\frac{p^2}{2}\left(5-3\left(\frac{p}{5}\right)\right)\pmod{p^3}.$$

These conjectured congruences are among the results proved in the paper, confirming the claims for all indicated positive integers and primes.