The packaging also offers some hints. It has a glass jar and a metal lid, which might click when you open it. My guess is, that bottle went through an oven before being shipped. Glass is heavy and expensive, which isn’t what the manufacturer wants. They wouldn’t use a packaging like this unless they really have to, which implies heat treatment.
If sterilized properly, that process killed all the microbes in it, which means you can safely ship it and store it in ambient temperatures for a very long time. Chemical reactions still take place, which might affect the taste if you store it for decades or centuries. However, if it’s properly sterilized, microbes aren’t involved in the process in any way. Once the lid is opened for the first time, airborne microbes immediately contaminate the contents and the clock starts ticking.
Salt and vinegar make the environment hostile for most microbes, but not all. This method will clearly help with limiting the growth rate, but keeping the jar in the fridge would be my advice as well. This way, you get the benefit of multiple methods of slowing down the growth rate.