Disclaimer: I am not a an expert/professional in the field.
For normal measures of general/emotional intelligence, the studies I found say that correlation is actually positive, e.g. the study "Distributed neural system for emotional intelligence revealed by lesion mapping", published in the apparently very reputable peer-reviewed journal Social Cognitive & Affective Neuroscience.
The study did not use WEIRD subjects and instead relied on Vietnam veterans who suffered brain damage in combat as participants. Other studies (relying on WEIRD subjects) found no relationship.
This article from the US National Library of Medicine gives a very good rundown of empirical studies about "gifted individuals"
For the “warmth” dimension describing socioemotional characteristics,
evidence favors great similarity between gifted and average-ability
individuals. The gifted are no more prone to depression, anxiety, or
suicide (Reis and Renzulli, 2004; Martin et al., 2010), show similar
levels of wellbeing and stress (Zeidner and Shani-Zinovich, 2011), and
are as agreeable as average-ability persons (Schilling, 2009; DeYoung,
2011), conscientiousness (Ackerman and Heggestad, 1997), and social
abilities (Schilling, 2009; overview: Neihart et al., 2002).
With regards to the subjects of the studies/books: Both Neihart et. al 2002 and Schilling 2009 talk about children/adolescents in Western democracies.
I haven't read Neihart, but the Amazon description mentions that "The book also summarizes several decades worth of research on special populations, including minority, learning-disabled, and gay and lesbian gifted students".