  The non-abelian Hodge correspondence maps a polystable
$\mathrm{SL}(2,\mathbb{R})$-Higgs bundle on a compact Riemann surface $X$ of
genus $g\geq2$ to a connection which, in some cases, is the holonomy of a
branched hyperbolic structure. On the other hand, Gaiotto's conformal limit
maps the same bundle to a partial oper, i.e., to a connection whose holonomy is
that of a branched complex projective structure compatible with $X$.
  In this article, we show how these are both instances of the same phenomenon:
the family of connections appearing in the conformal limit can be understood as
a family of complex projective structures, deforming the hyperbolic ones into
the ones compatible with $X$.
  We also show that, when the Higgs bundle has zero Toledo invariant, this
deformation is optimal, inducing a geodesic on Teichm\"uller's metric space.
