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Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh–Bénard convection (RBC) exhibits the classical 1 / 3 scaling as the Rayleigh number R a → ∞ with Prandtl number unity, consistent with Malkus–Howard’s marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin–Doering–Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same 5 / 12 scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the 1 / 3 scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.