On dimensional properties of topological products

For any n=2,3,…, there exist a metrizable compactum Φn and a compactum Yn such that and dim(Φn×Yn)=n+1<2n=ind(Φn×Yn). For any Tychonoff space X and any metrizable space Y, we have:(1) and(2) if, additionally, Y is strongly metrizable (in particular, strongly paracompact, or separable, or compact).