Is the identity $$\binom{p^k}{jp^{k-2}}\equiv \binom{p^2}{j} \;\mathrm{mod}\,p^3$$ true for all $k\geq 2$ and $j=1,\dots,p^2$ when $p=3$?
By Wolstenholme's theorem, we know it is true for all primes $p\geq 5$. Also, by Babbage's theorem, we know it is true for all primes if we replace modulo $p^3$ by $p^2$. However, I cannot find any proof or counterexample to this specific case with $p=3$. I have tried to apply Granville's result but I have not succeeded. Is there one? Is it actually true?