With the introduction of HTTP/3, whose transport is no longer the traditional TCP protocol but the novel QUIC protocol, research for solutions to the unfairness of Adaptive Streaming over HTTP (HAS) has become more challenging. That is, because of different transport layers, the HTTP/3 may not be available for some networks and the clients have to use HTTP/2 for their HAS applications instead. Therefore, the scenario that HAS over HTTP/3 (HAS/3) compete against HTTP/2 (HAS/2) must be considered seriously. However, there have been a shortage of investigations on the performance and the origin of the unfairness in such a cross-protocol scenario in order to produce proper solutions. Therefore, this paper provides a performance evaluation and root-cause analysis of the cross-protocol unfairness between HAS/3 and HAS/2. It is concluded that, due to differences in the congestion control mechanisms of QUIC and TCP, HAS/3 clients obtain larger congestion windows, thus requesting higher video bitrates than HAS/2. As the problem lies in the transport layer, existing client-side ABR-based solutions for the unfairness from the application layer may perform suboptimally for the cross-protocol case.