  We reassess the phase diagram of high-pressure solid hydrogen using
mean-field and many-body wave function based approaches to determine the nature
of phase III of solid hydrogen. To discover the best candidates for phase III,
density functional theory calculations within the meta-generalized gradient
approximation by means of the strongly constrained and appropriately normed
(SCAN) semilocal density functional are employed. We study eleven molecular
structures with different symmetries, which are the most competitive phases,
within the pressure range of 100 to 500~GPa. The SCAN phase diagram predicts
that the $C2/c-24$ and $P6_122-36$ structures are the best candidates for phase
III with an energy difference of less than 1~meV/atom. To verify the stability
of the competitive insulator structures of $C2/c-24$ and $P6_122-36$, we apply
the diffusion Monte Carlo (DMC) method to optimise the percentage $\alpha$ of
exact-exchange in the trial many-body wave function. We found that the
optimised $\alpha$ equals to $40 \%$, and denote the corresponding exchange and
correlation functional as PBE1. The energy gain with respect to the well-known
hybrid functional PBE0, where $\alpha = 25\%$, varies with density and
structure. The PBE1-DMC enthalpy-pressure phase diagram predicts that the
$P6_122-36$ structure is stable up to 210~GPa, where it transforms to the
$C2/c-24$. Hence, we predict that the phase III of high-pressure solid hydrogen
is polymorphic.
