Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels? A superchannel maps channels $\mathcal N:A\to B$ to channels $C\to D$ and admits a realization

 \begin{equation}
 \Theta(\mathcal N)
 =\mathcal D\circ(\mathcal N\otimes\operatorname{id}_S)\circ\mathcal E,
 \qquad
 \mathcal E:C\to A S,
 \quad \mathcal D:B S\to D.
\tag{1}
\end{equation} 
The memory system $S$ in Eq. (1) is internal to the superchannel. In a fully parallel $n$-use strategy, one applies $\Theta_i^{\otimes n}$ to a joint $n$-partite inserted channel and then tests the resulting output on a joint input state. A nested-adaptive strategy may instead recursively insert the channel produced by one tested use into the channel slot of another, with arbitrary compatible CPTP maps between uses and a final binary measurement. For $\mathsf S\in\{\mathrm{par},\mathrm{nest}\}$, let

 \begin{equation}
 \begin{aligned}
 \beta_{\varepsilon,n}^{\mathsf S}(\Theta_1\|\Theta_2)
 &:=\inf\{\beta_n(P):P\in\mathsf S_n,\ \alpha_n(P)\leq\varepsilon\},\\
 \zeta_{\mathsf S}(\Theta_1\|\Theta_2)
 &:=\lim_{\varepsilon\downarrow0}\liminf_{n\to\infty}
 -\frac1n\log_2
 \beta_{\varepsilon,n}^{\mathsf S}(\Theta_1\|\Theta_2),
 \end{aligned}
\tag{2}
\end{equation} 
where $\alpha_n$ and $\beta_n$ are the type-I and type-II errors. Is

 \begin{equation}
 \zeta_{\mathrm{nest}}(\Theta_1\|\Theta_2)
 =\zeta_{\mathrm{par}}(\Theta_1\|\Theta_2)
\tag{3}
\end{equation} 
for every pair $\Theta_1,\Theta_2$?
