Nayak and Yuen's rigidity conjecture for superdense coding

Let $(\tau,(W_i))$ be a $d$-dimensional superdense coding protocol. Let $\mathcal{H}_{A'}$, $\mathcal{H}_{A”}$, $\mathcal{H}_{B'}$, and $\mathcal{H}_{B”}$ be the Hilbert spaces in the asserted decomposition, with $\mathcal{H}_{B”}$ isomorphic to $\mathbb{C}^d$, and let $|\phi_d\rangle$ denote a maximally entangled state on $\mathcal{H}_{A”}\otimes\mathcal{H}_{B”}$. For operators $C,D,E$, write $C =_E D$ when $CEC^\dagger=DED^\dagger$. Let $[d^2]=\{1,\ldots,d^2\}$. Nayak and Yuen's rigidity conjecture. There exist a unitary operator $V$ on $\mathcal{H}_{A'}\otimes\mathcal{H}_{A”}$, unitaries $(C_i)_{i\in[d^2]}$ on $\mathcal{H}_{A'}$, an isometry $W:\mathcal{H}_B\to\mathcal{H}_{B'}\otimes\mathcal{H}_{B”}$, a density matrix $\rho$ on $\mathcal{H}_{A'}\otimes\mathcal{H}_{B'}$, pairwise orthogonal projectors $\{P_r\}$ on $\mathcal{H}_{A'}$ summing to the identity, and, for every $r$, an orthogonal unitary basis $\{E_{r,i}\}_{i\in[d^2]}$ for the space of $d\times d$ complex matrices, such that, with

$$\tau'=(V\otimes W)\tau(V\otimes W)^\dagger,$$

one has

$$\tau'=\rho^{A'B'}\otimes|\phi_d\rangle\!\langle\phi_d|^{A”B”}$$

and, for every $i\in[d^2]$,

$$(C_i^\dagger\otimes\mathbb{1})W_iV^\dagger =_{\tau'} \sum_rP_r\otimes E_{r,i}.$$

The conjecture asserts rigidity of every superdense coding protocol up to local isometries and the choice of an orthogonal unitary basis, with the auxiliary state permitting a shared random outcome $r$. The parser reports no resolution, so the conjecture remains open in the supplied source.