  We revisit the optimal heat transport problem for Rayleigh-B\'enard
convection in which a rigorous upper bound on the Nusselt number, $Nu$, is
sought as a function of the Rayleigh number $Ra$. Concentrating on the
2-dimensional problem with stress-free boundary conditions, we impose the full
heat equation as a constraint for the bound using a novel 2-dimensional
background approach thereby complementing the `wall-to-wall' approach of
Hassanzadeh \etal \,(\emph{J. Fluid Mech.} \textbf{751}, 627-662, 2014).
Imposing the same symmetry on the problem, we find correspondence with their
result for $Ra \leq Ra_c:=4468.8$ but, beyond that, the optimal fields
complexify to produce a higher bound. This bound approaches that by a
1-dimensional background field as the length of computational domain
$L\rightarrow\infty$. On lifting the imposed symmetry, the optimal
2-dimensional temperature background field reverts back to being 1-dimensional
giving the best bound $Nu\le 0.055Ra^{1/2}$ compared to $Nu \le 0.026Ra^{1/2}$
in the non-slip case. % We then show via an inductive bifurcation analysis that
imposing the full time-averaged Boussinesq equations as constraints (by
introducing 2-dimensional temperature {\em and} velocity background fields) is
also unable to lower this bound. This then exhausts the background approach for
the 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem with
the bound remaining stubbornly $Ra^{1/2}$ while data seems more to scale like
$Ra^{1/3}$ for large $Ra$. % Finally, we show that adding a velocity background
field to the formulation of Wen \etal\, (\emph{Phys. Rev. E.} \textbf{92},
043012, 2015), which is able to use an extra vorticity constraint due to the
stress-free condition to lower the bound to $ Nu \le O(Ra^{5/12})$, also fails
to improve the bound.
