  In this letter, we consider a positive geometry conjectured to encode the
loop integrand of four-point stress-energy correlators in planar
$\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space,
we characterize a positive subspace to which an $\ell$-loop geometry is
attached. The loop geometry then consists of $\ell$ lines in twistor space
satisfying positivity conditions among themselves and with respect to the base.
Consequently, the \textit{loop geometry} can be viewed as fibration over a
\textit{tree geometry}. The fibration naturally dissects the base into
chambers, in which the degree-$4 \ell$ loop form is unique and distinct for
each chamber. Interestingly, up to three loops, the chambers are simply
organized by the six ordering of $x^2_{12}x^2_{34}$, $x^2_{14}x^2_{23}$ and
$x^2_{13}x^2_{24}$. We explicitly verify our conjecture by computing the
loop-forms in terms of a basis of planar conformal integrals up to $\ell=3$,
which indeed yield correct loop integrands for the four-point correlator.
