The derived Hecke action formula for Taylor--Wiles primes

Let $E$ be the coefficient field, let $\mathcal{O}$ be its ring of integers, and let $\mathfrak{p}$ be a prime satisfying the conditions of the paper. Let $n\geqslant1$, let $q$ be a Taylor--Wiles prime of level $n$, and let $z\in(\mathcal{O}/\mathfrak{p}^n)\langle-1\rangle$. Let $\theta_q^{\vee}:k\langle-1\rangle\to\mathrm{U}_g^{\vee}\otimes k$ be the adjoint of the reduction map $\theta_q:\mathrm{U}_g\to k\langle1\rangle$, and let $\theta_q^{\vee}(z)^{\sim}$ be an arbitrary lift to $\mathrm{U}_g^{\vee}$. The derived Hecke action formula. There is an action $\star$ of $\mathrm{U}_g^{\vee}$ on $H^*(X_{\mathcal{O}[1/N]},\omega)[g]$, and there is $\alpha\in E$, such that

$$T_{q,z}\bar g=\alpha\overline{\left(\theta_q^{\vee}(z)^{\sim}\star g\right)}.$$

Here the bar denotes reduction modulo $\mathfrak{p}^n$. This conjecture identifies the derived Hecke operator with the action of the adjoint of the Taylor--Wiles reduction map, up to a scalar; the supplied material gives no evidence of resolution.