What happens to consumer surplus and profits as firms get more information in Hotelling's duopoly model? In the Duopoly on the line [0,1] with customers uniformly distributed, with firm A on the 0 side and firm B on 1's side, we know that if the firms have no information about the customer, there will be no price discrimination, and the market will be divided into segments [0, 0.5] and (0.5, 1] buying from A and B respectively. If the exact points at which the customers divide is not important, will notate this as [A, B].

If the firms know which half of the line the customer is from, then (loosely speaking) the profits decrease, as the Segments get partitioned not into [A, B], but into [A, B, A, B] with lots of competition going on. Consumer surplus also decreases as there is some inefficient buying.

But what happens when the information increases? Namely, the information of the segment that the customer is from increases from 2 in the previous case, to a large number N. At the limit, this is equivalent to perfect information. Would I be correct to guess that competition gets more fierce in every segment, and so both profits and consumer surplus decreases? Or will we revert back to [A, B]?

In the question above, I was looking for an intuitive answer, not one based on calculations and best responses. But what if I wanted to find the exact equilibrium price and profits at the limit (that is, the firms have perfect information about where the customer is from)?
The article Quality of Information and Oligopolistic Price
Discrimination by Liu and Serfes covers this topic in great detail. It also has a rather nice literature review.