Social media platforms like Twitter are extensively used during crises which seems to be a double-edged sword. On one hand, these platforms have an invaluable potential to become a reliable source of information and communication-tool for those involved in crisis situations. On the other hand, it is already known that current social media platforms allow misinformation to spread quickly - a fact that, in crisis situations, could widely jeopardize public safety. The reasons why misinformation spreads so quickly on Twitter in crisis situations can be traced back to several factors such as the platform’s design choices, as well as the user's lack of critical engagement with the information, but, as we argue in this article, also because of an inherent value hierarchy embedded in the design of the user interface. In this paper, we aim to first clarify what can be done effectively by Twitter to mitigate the spread of misinformation in crisis situations. By using a Value Sensitive Design framework, we argue that the current Twitter design promotes a multiplicity of values that are implemented in the user-interface design in ways that lead to conflicting effects on misinformation. We thus argue that a design intervention needs to change both the current value hierarchy and the design implementations of the values at stake. Our paper concludes with several design recommendations for elevating the user's critical engagement as well as a discussion about the moral implications of trying to streamline Twitter’s user interface design towards favouring truth-telling in situations of crisis.