I noticed that '2 of 8 URLs works', '/product-one/' and '/product-four/'. And also I noticed that there are Web Applications in IIS with same names - 'product-one' and 'product-four'. These apps are running the same application pool as web application that contains original page 'page.aspx'.
So I created six more web application in IIS with needed names ('product-two', 'product-three', etc.) and made them use corresponding application pool and pointed to new empty folder.
I know, this is not an elegant solution, but it works for me.