Need to test heterocedasticity before using the Kruskal-Wallis test

Question

My question is whether or not to test heterodasticity before using the Kruskal-Wallis test.

What I understand is that if I want to compare medians/mean, then I have to check for heterocedasticity, but if my goal is to use the test to check if one sample is stochastically different than the other, I don’t necessarily have to check for assumptions .

In the article by Kruskal and Wallis (Use of Ranks in One-Criterion Variance Analysis), at the end of page 17/40 or numbered 598:

4.2. Comparison of Means when Variability Differs Rigorously Interpreted, all we can conclude from a significant value of H is that the populations differ, not necessarily that the Means differ.

or among others

If you Wish to compare medians or Means, then the Kruskal-Wallis test also assumes that Observations in each group are identically and independently Distributed apart from Location. If you can Accept inference in Terms of dominance of one Distribution over Another, then there are Indeed no distributional assumptions.

I have already found teachers agreeing and disagreeing on this subject. I appreciate your guidance.