It may seem strange to use energy density in this context but in many ways those are even worse than batteries due to the massive amount of work they require and poor scalability because of all the physical infrastructure and space.
In fact, this exact scheme looks to have been posted to HN in 2014. From my understanding they are looking to re-purpose holes that have already been dug, for the most part, which is a good idea, but if the planned prototypes are only doing 250 kW nameplate with maybe 100 KWH of stored power, there is almost no way it's going to beat a battery plant you can have more or less just built to order already commercially. And said battery plants aren't even remotely good enough at purpose for meaningful energy storage, they're mostly used for what are considered "ancillary services" in the US power markets, like eating or producing reactive power, voltage support, etc, because the modern grid scale inverters can react to grid conditions faster than grid frequencies (50-60 Hz). More rarely they might be scaled big enough to shave the peak off a demand curve for 5-10 minutes to save transmission capacity or congestion.
Keep in mind all this in happening when LNG is basically free except for the cost of moving it, so you can also put in 1-60 MW combined cycle gas plants in short amounts of time with very well understood technology and proven manufacturers.