Conjecture on the optimal Toeplitz feedback scheme for the AWGN channel

Let the system be the AWGN channel with noisy feedback described in the paper, with blocklength $N$, feedback-noise variance $\sigma^2$, signal-to-noise parameter $\gamma$, power parameter $\rho$, strictly lower-triangular encoding matrix $\mathbf{F}$, and combining vector $\mathbf{q}$ subject to the power constraints in the source. Write $\\|\mathbf{F}\\|_F$ for the Frobenius norm. The received SNR is to be maximized over such $\mathbf{F}$ and $\mathbf{q}$. Optimal feedback-scheme conjecture. Given the power constraints, every maximizing pair

$$(\mathbf{F},\mathbf{q})$$

has the following form: $\mathbf{F}$ is a strictly lower-diagonal Toeplitz matrix, meaning that all entries on each lower diagonal are equal; \

(1+\sigma^2)\\|\mathbf{F}\\|_F^2=N\gamma\rho; \

for some $\beta\in(0,1)$, \

\mathbf{q}=\sqrt{\frac{1-\beta^2}{1-\beta^{2N}}}\left[1,\beta,\beta^2,\ldots,\beta^{N-1}\right]^T. \

The claim is empirical: it is motivated by iterative optimization from different initial vectors and parameter values, together with unsuccessful random searches for a better form; no proof or resolution is supplied in the source.