Question

in most cases would a soil sample from temperate deciduous forest be more or less nutrient-rich than soil sample from tropical rain forest

Answer 1

A soil sample from a temperate deciduous forest would be more nutrient rich that a sample from the tropical rain forest.
In the rain forest the tree tops form a canopy and the tops of trees absorb 90% of the sunlight and only 10% can get down to the understory which is the area under the canopy. The soil is very nutrient poor.
In a temperate deciduous forest there is no canopy so more sunlight can get to the soil. Which makes it more nutrient.