There is no known conflicts with BD. I assisted other with different issues that were using both with no issue. I am familiar with this site and as far as I can remember Connections is being displayed exactly where it should be using the Slim Plus template. All the individual city pages have the Connections directory embedded on them with the [connections] shortcode.
That said… I actually do not see any where on the site where a link is setup to use BD… though a random guess I found this page which does use BD and not Connections and I do not see Connections interferring with BD and I do not see any Google Maps at all in either plugin. So everything seems in order. If you can provide some direct links of where there is a issue/conflict, I’ll take another.