The short answer is “most likely”. Having spent the last 15 years in Brand Protection (anti-counterfeiting) I learned quite a few ways of fighting counterfeits. My actions caused the closing of two factories and the dreaded “third strike” to another which stopped them from producing counterfeits.
Here are some of the things that helped me:

An excellent attorney: I hired a young Chinese attorney who had just received a master’s degree in Intellectual Property from the U of Illinois. I gave him some resources and told him to figure out a way to go after the counterfeiters. He developed an overall plan that included working with the Chinese Customs Agency. In a meeting with the US Customs the Chinese pointed to the program as a best practice that other companies should follow.
Data: I can’t stress enough how important it is to have trade data for the countries where the problem exists. It points out the importers involved as well as the exporters supplying the counterfeits.
Customs Agencies: If the countries have brand registration (both China and the US, have it) you can give a heads up to Customs Agencies to be on the lookout for counterfeits. They have software that helps them review shipments. Less than 5% of containers are inspected.
Chinese staff: I set up staff in China, and they analyzed the export data, identified targets, and accompanied me on market visits.
Knowing the hot spots: In China there are two places you should go to see if your products are being counterfeited: The Yiwu market in Yiwu and the Canton Fair in Guangzhou. Outside of China you have Dubai and India.
The service department: This is usually the first place you will hear of fake products as they are sent in for service by unsuspecting buyers. If I was running a gang of thieves I would buy products in a store, replace them with fake products, and then return them to the stores for a refund and sell the originals online. If you have DIF policies with your retailers, you at least want to spot check the returns.

An aggressive approach would be for you to make your own cheap products to compete with the counterfeits and sell them in places like the Yiwu market and markets where you see counterfeits. You work with manufacturing to make a cheaper version and use that to compete with any counterfeits.
My worst case: A market where I had close to a 100% share was 20% in five years despite an aggressive anti-counterfeit campaign where over half a million dollars’ worth of counterfeits were confiscated and destroyed.
Happy Thanksgiving to those of you in the USA.
Bob Thomas