Supercongruences involving products of two binomial coefficients modulo $p^4$

In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$\sum _{k=(p+1)\mathrm{/}2}^{p-1}\frac{{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}{k{16}^k}\equiv -\frac{21}{2}{H}_{p-1}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^4),$$ where $H_n$ denotes the $n$-th harmonic number.