= Solution
Write $N=2k+1$ and abbreviate the <Dicke states> by $|D_j^n\rangle=|\Psi_j^n\rangle$. Alice's shared resource has decomposition
$$
\frac1{\sqrt2}\left(|0\rangle|D_{k+1}^N\rangle+|1\rangle|D_k^N\rangle\right).
$$
Follow the actual circuit order: a <controlled-NOT gate> from the input to $q_0$, a <Hadamard gate> on the input and a <Pauli X gate> on $q_0$, followed by <computational basis> measurements. Just before measurement, the joint <vector> is
$$
\begin{aligned}
\frac12\big[&|00\rangle(\alpha|D_k^N\rangle+\beta|D_{k+1}^N\rangle)\\
+&|01\rangle(\alpha|D_{k+1}^N\rangle+\beta|D_k^N\rangle)\\
+&|10\rangle(\alpha|D_k^N\rangle-\beta|D_{k+1}^N\rangle)\\
+&|11\rangle(\alpha|D_{k+1}^N\rangle-\beta|D_k^N\rangle)\big].
\end{aligned}
$$
The <Dicke states> in each branch are <orthogonal>, so each branch has squared <norm> $1/4$. Conditioning on outcome $00$ and normalizing therefore gives the state of the other $N$ parties:
$$
\boxed{|\Omega\rangle=\alpha|D_{(N-1)/2}^N\rangle+\beta|D_{(N+1)/2}^N\rangle.}
$$
This is the encoding used in <Dicke-resource telecloning>. The extra <Pauli X gate> on Alice's resource <qubit> is essential to the displayed ordering of the amplitudes.
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