Sun's integrality conjectures for central-binomial sums

Let $S_k$ and $S_k^{(2)}$ be the sequences defined by the corresponding central-binomial sums in the source. Sun's integrality conjectures. For every positive integer $n$, one has

$$\frac{4}{n^2}\sum_{k=0}^{n-1}kS_k\in\mathbb{Z},\qquad \frac{1}{n^2}\sum_{k=0}^{n-1}S_k^{(2)}\in\mathbb{Z}.$$

These conjectures of Z.-W. Sun are included among the congruences proved in the paper, so they are resolved.