For $p\in \mathbf{N}$ is $$p^3 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$?

Notice that the second condition is equvalent to say: "$p$ is a prime or $p$ is a Poulet number". Or

is $$p^4 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$? Notice that the first condition here is equvalent to say: "$p$ is a Wolstenholme number"