I guess submenu is a module - Modules are displayed in specific pages, by assigning them on menu items.
It's possible that you don't have a menu-item for the Product Details Page - if you do, then you need to assign the module to this menu-item as well.
It might also worth trying to use the "exclude assignment" option for your module:
Instead of using "show on the selected menu items", use "show on all menu items except the selected". Of course, you will have to select all the menu items that you don't want the module to appear.
Other things that could help:
Check to see, if Joomshopping is giving you options for using specific menu-item id for the product details page. If something like this exist, make sure you have a menu item created (even hidden) to hold the product-details page, and assign on this menu-item the modules you want to show.
Finally, the Advanced Module Manager by NoNumber extends the core Moduler Manager, providing many more options for Module assignments, which may come handy in your case.