It will as they asymptotically approach the raw material costs. (What that point is I don't know, but there is a floor to cell costs)
The raw material cost depends on the state of technology. As density increases less raw material is needed per kWh. Also the specific mix of materials changes. For example, while the price of cobalt was high, Tesla was making serious progress in reducing the amount of cobalt per kWh. Now that it is cheap again, perhaps this is not so important. The technology advances at least have option value. For example, if cobalt were to go sky high in price, Tesla would have an option to switch to lower cobalt chemistry. So over time the advance of the tech enable both the cost of raw material per kWh to decline and to navigate volatility in the price of specific minerals.
We also have the possibility that certain minerals may actually become cheap to mine as their industry advance to higher scale production. Again the cobalt market has historically be too small to motivate miners to specifically target it. Rather they target copper or nickel which are have much bigger markets, and this affords some associated production of cobalt. But as the cobalt market increases in value, miners will become more competitive in bringing cobalt to market. Lithium has huge potential for technology to decrease mining costs. For example, ocean water could be a good source of lithium if you had a cheap technology for separating trace amounts of the mineral from huge volumes of water. The bigger the lithium market becomes, the more R&D spending goes to improving extraction technologies, which in turn can lower the cost of the raw material.
So the way I think about the experience curve is that it works on every segment of the supply chain. As the production of batteries scales up, it forces every supplied component in the production chain also to scale up. As any one of these supply components scales up, it too is subject to some experience curve, albeit the learning rates may be different and it may take longer to double scale of a particular component. Nevertheless, as each component is scaling into its experience curve, each component may arrive at lower costs. This in turn enables the downstream product, batteries, to continue down its experience curve too cutting total cost. This may sound pretty optimistic, but it is also what makes incumbent technologies so hard to beat. Incumbent technologies have huge well optimized supply chains. The lithium battery supply chain is nowhere near as optimized as say the supply chain for internal combustion engines. But it takes ICE a long time to double cumulative production so cost declines over time are slower and more easily overwhelmed by inflationary forces in the general economy. But if you look at all the components of ICE, those supply streams are large, fiercely competitive and highly optimized. The cumulative effect is hard for any radically new technology to beat. But as batteries scale, the whole supply chain for batteries will become large, fiercely competitive, and highly optimized too.
Another issue particular to the battery supply chain is that most of the minerals can be recycled. So as the tonnage of used lithium batteries grows, recycling becomes an alternative supply to mining. The logistics, scale and technology of recycling batteries will improve and follow an experience curve of its own. The more efficient recycling becomes, the greater the salvage value of batteries will be. So even if over the long run the mineral cost of batteries dominates and slows down the experience curve price reductions for gross battery pack cost, the net cost (new cost minus salvage value) can continue to decline. One is essentially paying rent on the salvage value of the battery plus a small price for recycling and re-manufacturing batteries. The faction of raw materials (not obtained by recycling) can become exceedingly small once EVs have replaced nearly all ICE vehicles in use. Indeed for a real mind bender, consider the possibility that density doubles in ten years. This suggests the possibility that the material used for 1 kWh today could be recycled in about ten years to produce a battery with nearly 2 kWh capacity. The only slippage here is changes in the ratios of minerals and some loss of materials in recycling. Even so the demand for raw materials needed to double battery production over a decade is relatively small. The problem is that right now demand EV batteries will double 5 to 7 times over the next ten years, while doubling density will only suffice for 1 of those doublings. So for the next ten or so years the EV industry will need massive inputs from mining while recycling is only a small contributor to battery supply. But going out 20 years or more, recycling becomes the dominant supply. This is one of the ways that the battery economy will be substantially different from the oil economy. Oil is for the most part a single use mineral, while batteries are highly recyclable. The oil economy was easy to ramp up, but is too costly to maintain (including climate change). However costly the battery economy is to ramp up, it will be much cheaper to maintain.