Galatius--Venkatesh compatibility of derived Hecke actions

Let $K$ be the level, let $\mathfrak m$ be the maximal ideal associated to $\pi$, and let

$$v\in H^1_f\left(\mathbb Z\left[\frac1S\right],\operatorname{Lie}\check{\mathrm G}\right)^\vee, \qquad v^*\in H^1_f\left(\mathbb Z\left[\frac1S\right],\operatorname{Lie}\check{\mathrm G}\right),$$

with $f\in H^*(Y(K),E)_\mathfrak m$. Galatius--Venkatesh compatibility conjecture. The degree-raising and degree-lowering actions satisfy

$$v^*\!\left(v.f\right)+v\!\left(v^*.f\right)=\langle v^*,v\rangle\,f.$$

This predicts compatibility between the derived Hecke action constructed in the paper and the action of Hansen and Thorne, under the smoothness assumption on the ordinary deformation ring.