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The airline industry was severely hit by the COVID-19 crisis with an average demand decrease of about $64\%$ (IATA, April 2020) which triggered already several bankruptcies of airline companies all over the world. While the robustness of the world airline network (WAN) was mostly studied as an homogeneous network, we introduce a new tool for analyzing the impact of a company failure: the `airline company network' where two airlines are connected if they share at least one route segment. Using this tool, we observe that the failure of companies well connected with others has the largest impact on the connectivity of the WAN. We then explore how the global demand reduction affects airlines differently, and provide an analysis of different scenarios if its stays low and does not come back to its pre-crisis level. Using traffic data from the Official Aviation Guide (OAG) and simple assumptions about customer's airline choice strategies, we find that the local effective demand can be much lower than the average one, especially for companies that are not monopolistic and share their segments with larger companies. Even if the average demand comes back to $60\%$ of the total capacity, we find that between $46\%$ and $59\%$ of the companies could experience a reduction of more than $50\%$ of their traffic, depending on the type of competitive advantage that drives customer's airline choice. These results highlight how the complex competitive structure of the WAN weakens its robustness when facing such a large crisis.