Can you raise a city center or district above level 2?

Question

The user's manual states that you can level up a city center or district by surrounding it with at least four other districts.

[A district tile] can be upgraded by surrounding it with 4 or more other District tiles. . . . If a District (the city center is also considered as a district) becomes surrounded by 4 other District tiles, its efficiency is greatly increased.

But the Borough Streets expansion's "per level" city tile bonuses imply that you can go further:

• +1 Dust on city tile
• +2 Dust per level on city tile

However, hovering over level 1 and 2 district tiles on the map contradicts the above:

• Level 1: +1 Dust from City
• Level 2: +3 Dust from City

The same bonuses apply and show up on the map for both Science and Influence. I assume the "per level" actually means "per additional level" giving +1 from level 1 and +2 from level 2. But that implies higher district levels even more.

Can a district reach level 3 and beyond?

Answer 1

As of 2014-11-09, it is not possible to upgrade a district or city center beyond level 2. I've tried the Vaulters and the Broken Lords in several cities each, but without success, and I don't expect other playable factions to behave differently. By the end of my games a missing science breakthrough can't have been an issue as well because everything except the win conditions in the last circle was researched.

Here are some city layouts that should work but don't (the expected 3's came out as 2's every time):

     1 2 1          1 2 1               1 2 1
    1 2 2 1        2 2 2 2             2 3 3 2
       3      or  1 2 3 2 1  or even  1 2 4 2 1
    1 2 2 1        1     1             2 3 3 2
     1 2 1                              1 2 1

It doesn't matter if the expected level 3 district is the city center or a built district.

Edit: I have verified (thanks to the anonymous suggestion) that the Cultists faction can have level 3 districs and city center. One turn after capturing their city with the Vaulters, all their level 3 districts downgraded to level 2. Maybe this is on purpose, after all the Cultists are limited to just one city.

Edit: I have verified playing the Cultists that their districts can reach level 3 but not higher. My city had several level 3 districts, none of them would rise to level 4, though.

Answer 2

Yes. Looking at a district, take the 4 highest level districts that it borders. Its level is equal to one more than the smallest of those 4. That is, in order to get a level 3 district you must border it with at least 4 level 2 (or greater) districts.