According to the Polarity Hypothesis, the presence or absence of a processing cost for Scalar Inferences (SIs) depends on their polarity. This hypothesis predicts, among other things, that the processing of lower-bounding SIs should not be affected by cognitive load the same way upper-bounding SIs are. To date, evidence in support of this prediction comes from the comparison between upper-bounding and lower-bounding SIs elicited by disparate scalar words. In this paper, we report on two dual-task experiments testing this prediction in a more controlled way by comparing upper-bounding and lower-bounding SIs arising from the same scalar words or scale-mates operating over the same dimension. Results show that, for these more minimal comparisons, lower-bounding SIs involve comparable cognitive demands as their upper-bounding counterparts. These findings challenge the idea that load effects are consistently modulated by SI polarity and suggest instead that these effects are relatively consistent across different types of SIs.