Good morning!
Great question, and a little history on this may be helpful.
In 2015, Google drew attention to the fact that users were increasingly using "near me" as a modifier for searches, and in an effort to prompt local results. According to Google the use of such terms as "near me" and "nearby" had doubled within the foregoing year. As a result of this, the use of "near me" terms became the subject of optimization experiments.
However, fast forward a couple of years, and Google came out with an update on this topic, which you can read in full here: https://www.thinkwithgoogle.com/consumer-insights/local-search-mobile-search-micro-moments/ To sum up, these near-type searches had begun to subside. The obvious answer to the "why" of this is that users are becoming increasingly aware of that fact that Google will automatically localize a huge variety of searches to which they assign a local intent, without the user having to modify their search at all, with "near me" or even with a city name. You look up "pizza" on your phone in downtown San Francisco and you can be pretty sure Google is going to show you pizza places nearest you without you having to add a modifier of any kind.
So, does this mean that brands should no longer be including terms like "near me" in their website optimization? No, it doesn't mean that, as, according to Google, people are still searching this way. It's just that fewer of them are, so whereas in 2015 SEOs might have advised quite a good sprinkling of near-type language in a website's tags and text, in 2018, the recommended sprinkling would likely be quite a bit less. And I'll add an it-depends on this, too. There could be some demographics, perhaps by region, or age, or level of technological sophistication, or even business type where use of "near me" could still be quite prevalent. And let's not forget about voice search, and whether more natural language patterns might be leading folks to be asking their assistants, "Where's the best pizza near me?" instead of just saying "pizza".
Which brings me to why I'm voting in major favor of your A/B testing! It will help you identify which language best matches your customers' style of search. It would be really nice if, after your test, you might come back to this thread and let the community know how it went. Good luck!