In recent years, an unprecedented growth in the usage of mobile devices for web browsing poses a challenge for the service providers to assure the user-perceived quality. In the context of web quality of experience (QoE), quality perception is mostly dominated by the page load time (PLT). HTTP/2 protocol, with the server push feature, promises to address the design limitations of HTTP/1.1 that inhibit optimal web performance. However, it remains largely unclear if HTTP/2 can really improve web QoE for mobile browsing. In this paper, we experimentally investigate the web QoE with HTTP/2. We assess the web QoE for several popular websites on a controlled testbed emulated with real 4G/LTE and 3G network traces. Our experiments investigate the impact of both network latency and packet loss ratio on the mobile web QoE. The results clearly show 24% improvement in the PLT, on an average, with HTTP/2 over mobile networks. However, we identify that HTTP/2 with server push is necessarily not the fail-safe solution for improving mobile web QoE under all conditions. We noticed that HTTP/2 loads the web pages slower than HTTP/1.1 when the network packet loss ratio is more than 2%. Our study could be used as the basis to derive a set of guidelines on the usage of the HTTP/2 server push to improve the end-user web QoE, especially in mobile devices.