Much of the answer has to do with what is called casinghead gas. (Aka “associated” gas.) That is, natural gas produced as part of oil drilling operations, and separated at the casing head. I do not know of good national figures for this, but Texas keeps good records. You can see from the below chart how natural gas production has declined with low prices, but the boom in unconventional oil has caused casinghead gas production to skyrocket. The end result has been stability in total output despite the low prices:
What happens if oil prices stay low for a prolonged period of time? New oil drilling will fall off — this has already happened. If prices remain low for more than a year or two, then production oil will fall. Associated gas will fall along with it. And that will drive gas prices up.
In short, a prolonged period of low oil prices will cause natural gas prices to rise.
The problem is that all this operates with long and uncertain lags. Which means that I cannot predict when prices will rise or how long they will remain high. What I can predict is that low oil prices mean more volatility in the natural gas market.
Buckle your seat belts!