I'm not wondering about the effect culture has on conformity (the territory), I'm wondering about the effect culture has on my prediction of conformity (the map). ... Is their map any different from mine?

Notice that their territory is different from yours. Just that would make you expect their map to be different.

One question that you may ask is whether the bias (the difference between the territory and the map) is a function of the territory: do people in collectivist cultures mis-estimate the prevalent conformity in a different way from people in individualist cultures?

I don't think this is a useless question.

It is not. Consider, for example, one of the issues in political studies: why repressive regimes which present a solid and impenetrable facade tend to collapse very rapidly when the first cracks in the facade appear? One of the answers is that it's a consequence of available information: a lot of people might be very unhappy with the regime but as long as they believe that they are a powerless minority they will hide and do nothing. The first cracks basically tell these people "you're not alone, there are many of you*, and the regime collapse follows soon thereafter.

Note the parallels to estimating the conformity of other people.