  The super-Planckian features of radiative heat transfer in the near-field are
known to depend strongly on both material and geometric design properties.
However, the relative importance and interplay of these two facets, and the
degree to which they can be used to ultimately control energy flow, remains an
open question. Recently derived bounds suggest that enhancements as large as
$|\chi|^4 \lambda^{2} / \left(\left(4\pi\right)^{2} \Im\left[\chi\right]^{2}
d^{2}\right)$ are possible between extended structures (compared to blackbody);
but neither geometries reaching this bound, nor designs revealing the predicted
material ($\chi$) scaling, have been previously reported. Here, exploiting
inverse techniques, in combination with fast computational approaches enabled
by the low-rank properties of elliptic operators for disjoint bodies, we
investigate this relation between material and geometry on an enlarged space
structures. Crucially, we find that the material proportionality given above
does indeed emerge in realistic structures. In reaching this result, we also
show that (in two dimensions) lossy metals such as tungsten, typically
considered to be poor candidate materials for strongly enhancing heat transfer
in the near infrared, can be structured to selectively realize flux rates that
come within $50\%$ of those exhibited by an ideal pair of resonant lossless
metals for separations as small as $2\%$ of a tunable design wavelength.
