On two conjectural supercongruences of Apagodu and Zeilberger

Let the numbers ${\alpha }_n,{\beta }_n$ and ${\gamma }_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad$\text{and}$\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime $p\ge 5$ and positive integer $n$ \begin{align*} \alpha_{np}&\equiv \left(\frac{p}{3}\right) \alpha_n \pmod{p^2},\ \beta_{np}&\equiv \begin{cases} \displaystyle \beta_n \pmod{p^2},\quad &$\text{if }p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$,\ -\gamma_n \pmod{p^2},\quad &$\text{if }p\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$, \end{cases} \end{align*} where $\left(\frac{\cdot }{p}\right)$ denotes the Legendre symbol. These two supercongruences were recently conjectured by Apagodu and Zeilberger.