Equality of modified and primitive Kato zeta-element modules

Let $T_f$ be a Galois-stable $\mathcal{O}_\pi$-lattice of $V_f$ satisfying Assumption (Im0), and let $r\in\mathbb{Z}$. Assume that $p$ does not divide $N$. For $i=0,\ldots,p-2$, let $\Lambda^{(i)}$ denote the relevant isotypic Iwasawa algebra, and let $\mathbf{z}_{\mathbb{Q},\delta_{f,\mathrm{cris}}}(f,i,k-r)$ and $\mathbf{z}_{\mathbb{Q},\gamma_0}(f,i,k-r)$ be the corresponding zeta elements in $\mathbb{H}^1(j_*T_{f,i}(k-r))$. Equality of zeta-element modules. For every $i=0,\ldots,p-2$,

$$\Lambda^{(i)}\mathbf{z}_{\mathbb{Q},\delta_{f,\mathrm{cris}}}(f,i,k-r)=\Lambda^{(i)}\mathbf{z}_{\mathbb{Q},\gamma_0}(f,i,k-r)$$

in $\mathbb{H}^1(j_*T_{f,i}(k-r))$. The authors present this as a question motivated by the role of optimal periods in the formulation of the Iwasawa main conjecture; no resolution is given.