The Sharifi-inspired Hecke annihilation conjecture for $\varpi$ away from $p$

Let

$$\varpi:H_1(X_1(N),C_{\Gamma_1(N)}^0,\mathbf{Z}_p)\longrightarrow K_2\left(\mathbf{Z}[\zeta_N,\frac{1}{Np}]\right)\otimes_{\mathbf{Z}}\mathbf{Z}_p$$

be the homomorphism induced by the Manin-symbol map. Let $T_n$ denote the Hecke operators and let $\langle d\rangle$ denote the $d$th diamond operator. Sharifi-inspired away-from-$p$ Hecke conjecture. The map $\varpi$ is annihilated by

$$T_n-\sum_{\substack{d\mid n\gcd(d,N)=1}}\frac{n}{d}\cdot\langle d\rangle$$

for all $n\ge1$ such that $\gcd(n,p)=1$. This conjecture is inspired by Sharifi's work, but the source says that the present level situation is not covered by the cited Sharifi and Fukaya--Kato settings; no resolution is given.