On a conjectural congruence of Guo

Let $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ be a prime and $r$ a positive integer. We show that $$\prod _{k=1}^{(p^{2r}-1)\mathrm{/}2}\frac{4k-1}{4k+1}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)\mathrm{.}$$ This confirms a recent conjecture of Guo.