In imprecise probability theories, independence modeling and computational tractability are two important issues. The former is essential to work with multiple variables and multivariate spaces, while the latter is essential in practical applications. When using lower probabilities to model uncertainty about the value assumed by a variable, satisfying the property of 2-monotonicity decreases the computational burden of inference, hence answering the latter issue. In a first part, this paper investigates whether the joint uncertainty obtained by main existing notions of independence preserve the 2-monotonicity of marginal models. It is shown that it is usually not the case, except for the formal extension of random set independence to 2-monotone lower probabilities. The second part of the paper explores the properties and interests of this extension within the setting of lower probabilities.