Checking for stationarity in LSTM
In principle no you do not need to check for stationarity nor correct for it when you are using an LSTM.

The thing about stationarity is that it makes prediction tasks much more efficient, and stable. Think about stationarity in terms of a target. When what you are trying to predict is not stationary, it is like trying to shoot a moving target. It isn't impossible to do, but there is a better way to do things.

If you are trying to predict a stationary variable, then what you are trying to do is hit a non-moving, for lack of a better word, stationary target. It is staying in the same spot relative to you.

Correcting for non-stationarities in an ARIMA model is necessary because it is one of the fundamental assumptions that you make about that particular model. LSTMs do not make that assumption. So you don't have to do it with an LSTM. Seems like the answer to your question is the same as what the people you asked originally. Case closed, right?

Well... you remember shooting at moving targets is much more difficult than shooting at a stationary target right? You may want to force stationarity on a problem that you are going to run against an LSTM, simply because you may be able to eek out a little bit more performance by making it easier for the neural network to learn. So although you don't need to do it, it may still be a good idea and give you a boost in performance.