Conforming schemes directly compute guaranteed upper bounds on the exact eigenvalues by the min-max principle and the rigorous estimation requires access to guaranteed lower eigenvalue bounds (GLB). This talk discusses the verifiable assumptions on the maximal mesh-size and discretisation parameters that result in direct GLB for 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. Both schemes come with an a priori quasi-best approximation property and allow a 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.