On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:

$\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; 4$

This fact is for instance mentioned in the comments of this question, but I have been unable to find a proof of it.

My question is: Is it true that on an 8-manifold, the analogous relation

$\mathfrak{P}(w_4) \stackrel{?}{=} p_2 \; {\rm mod} \; 4$

holds?