If you had asked me whether I see more potential for battery prices to go down than turbine prices I would 100% say that batteries have more potential. Why?

1. Because there is a lot of pressure for them to go down, in literally every form of technology, be it phones, robots, drones, EVs, ... Turbines are also important, but they have had that pressure for decades and are relatively finished. Any further improvement will give diminishing results outside of specialized applications and demands

2. Because the technology for batteries is relatively open ended in comparison. Turbine prices are mainly about how that steel can be precision engineered and while there is some innovation at this front, the gains from that innovation have slowed for a while now. With batteries however the race is still very open and it would not be naive to assume some new chemistry will be found that reduces the cost and energy density drastically.

I don't know the economics around precision engineering well enough to intelligently how high demand impacts the manufacturing cost, but I could imagine it behaving slightly different than the cost of simpler processes, especially since good CNC machines are hefty investments.

It's a different kind of manufacturing.

China leads the world in battery manufacturing, it's not really so hard in an absolute sense, but gee do you have to get the cost down and get the quality up so you don't have batteries blow up because of manufacturing defects.

To make gas turbines though you need to get the last bit of performance available out of metals, manufacture parts with punishing tolerances, and do it with high quality so turbines don't blow up because of manufacturing defects. At this point in time China cannot make competitive gas turbines. China's COMAC C919 airliner is 100% dependent on western engines for instance.