Experimental evidence for the Gardner transition, theoretically predicted to arise deep in the glassy state of matter, is scarce. At this transition, the energy landscape sensed by the particles forming the glass is expected to become more complex. In the present Letter, we report the dielectric response of two typical glass formers with well-pronounced Johari-Goldstein β relaxation, following this response down to unprecedented low temperatures, far below the glass transition. As the Johari-Goldstein process is believed to arise from the local structure of the energy landscape, its investigation seems an ideal tool to seek evidence for the Gardner transition. Indeed, we find an unusual broadening of the β relaxation below about 110 K for sorbitol and 100 K for xylitol, in excess of the expected broadening arising from a distribution of energy barriers. These results are well consistent with the presence of the Gardner transition in canonical structural glass formers.