In human–robot collaboration, people's perceptions of robot abilities can shape how they interact with it. Drawing on insights from a pilot study, we hypothesized that a robot's backstory and gaze patterns might strengthen perceptions of competence. To examine this, we conducted a 2×2 (backstory x gaze patterns) mixed-method study in a museum-based lab setting, where 66 participants completed two open-ended gift-box assembly tasks (makeup vs. pet-owner items) with a Misty robot. Contrary to our expectations and much of the HRI literature, we found no significant differences across experimental conditions. However, participants' interactions revealed important insights about collaboration dynamics. Two distinct collaboration levels emerged: while a quarter of participants maintained low levels of collaboration, the majority engaged in highly collaborative exchanges with the robot. These collaboration levels were associated with different perceptions of warmth and competence and different interaction patterns. A thematic analysis showed that in low-collaboration cases, participants' self-efficacy (i.e. the confidence they held in their own task competence) did not appear to influence the interaction. However, within high-collaboration cases, self-efficacy shaped how participants engaged with the robot. Those with high self-efficacy critically evaluated the robot's contributions against their own expertise, whereas those with low self-efficacy leaned more heavily on the robot's guidance. Across both groups, several factors seemed to enhance collaboration: the robot's answer justifications, occasional pushback from the robot, and participants' growing familiarity with the interaction. We discuss how future systems should account for users' confidence levels, balance responsiveness with constructive disagreement, and support adaptation to foster more balanced and effective collaboration.