  In the past few years, the unifying frameworks of 4-dimensional Chern-Simons
theory and affine Gaudin models have allowed for the systematic construction of
a large family of integrable $\sigma$-models. These models depend on the data
of a Riemann surface $C$ (here of genus 0 or 1) and of a meromorphic 1-form
$\omega$ on $C$, which encodes the geometry of their spectral parameter and the
analytic structure of their Lax connection. The main subject of this paper is
the renormalisation of these theories and in particular two conjectures
describing their 1-loop RG-flow in terms of the 1-form $\omega$. These
conjectures were put forward in [2010.07879] and [2106.09781] and were proven
in a variety of cases. After extending the proposal of [2010.07879] to the
elliptic setup (with $C$ of genus 1), we establish the equivalence of these two
conjectures and discuss some of their applications. Moreover, we check their
veracity on an explicit example, namely an integrable elliptic deformation of
the Principal Chiral Model on $\text{SL}_N(\mathbb{R})$.
