When interest rates are lower than they have ever been in the past, the price discrepancy between a long-term European put option and an otherwise identical short-term put option will be the greatest. European options can only be exercised at their expiration date. Therefore, a longer time to expiration implies that the option holder will have to wait longer to receive money from the sale of the underlying asset. This waiting period represents a lost interest opportunity. When interest rates are low, this lost interest is reduced, making the long-term option more valuable compared to the short-term option. Therefore, in a low-interest-rate environment, the price discrepancy between the long-term and short-term options will be the greatest.

Choice A is incorrect. While it's true that lower volatility would decrease the price of both options, it wouldn't necessarily increase the price discrepancy between them. The impact of volatility on option pricing is more complex and depends on other factors such as the strike price and time to maturity.

Choice C is incorrect. Higher interest rates could potentially increase the price discrepancy between long-term and short-term options, but not necessarily to a greater extent than extremely low interest rates. The relationship between interest rates and option prices isn't linear, so we can't say definitively that higher rates will lead to a larger price discrepancy.

Choice D is incorrect. As explained above, neither low market volatility nor high interest rates would necessarily lead to a greater price discrepancy between long-term and short-term options than extremely low interest rates would.