Segmentation analysis: Correct test to use?

As part of a larger study, we have collected a wealth of data on the interactions customers engage in when buying and using a service.

We have tried to look at this relatively close to reality. Hence, we distinguish 8 phases that tend to - but are not necessarily – sequential, 12 different interaction channels, and 9 different means of interaction (how the interaction happened, e.g., via telephone or online). When considering all these options, basically no two journeys are the same, which calls for a meaningful segmentation of customers with regards to their interaction preferences.

Based on a large qualitative study we ran on this exact topic as well as the results of descriptive statistics, we have identified a segmentation methodology that distinguishes 4 segments based on customer orientations/goals.

What we would like to prove is that this is a decent way to segment customers (from descriptives we know that it is better than observable behavior and demographics). But what would be the best statistical test to do so?

Thanks so much for any recommendations!