A new class of skeletal methods provides direct guaranteed lower eigenvalue bounds (GLB) under verifiable assumptions on the maximal mesh-size and discretisation parameters. The verification of the GLB condition requires the knowledge of some stability constants and its validity implies that the computed discrete eigenvalue is already a GLB. This talk discusses the hybrid-high order (HHO) eigenvalue solver of Carstensen-Ern-Puttkammer [Numer. Math. 149, 2021] and its recent modification with an even simpler $p$-robust parameter selection. We prove an a priori quasi-best approximation property and establish stabilization-free reliable and efficient a posteriori error control. Computer benchmarks provide striking numerical evidence for optimal high-order convergence rates of the associated adaptive mesh-refining algorithm.