Time-average code-length conjecture for zero-delay LQG quantizer coding

Let $g$ be a bijection as described in the cited power-law or exponential-envelope theorem. At time $\mathsf{t}$, the lossless encoder computes

$$\overline{\boldsymbol{\mathbf{\mathsf{q}}}}_{\mathsf{t}}=g({\boldsymbol{\mathbf{\MakeLowercase{q}}}}_{\mathsf{t}})$$

and encodes it using

$${\boldsymbol{\mathbf{\MakeLowercase{a}}}}_{\mathsf{t}}=C_{\mathsf{t},\overline{{\boldsymbol{\mathbf{\mathsf{{q}}}}}}^{\mathsf{t}-1}}(\overline{{\boldsymbol{\mathbf{\mathsf{{q}}}}}}_{\mathsf{t}}),$$

where the encoding function is constructed via the cited exponential-envelope scheme when $\mathsf{m}\leq 2$, or via the cited scheme with the zero-delay modification using Shannon–Fano–Elias coding otherwise. The encoding is prefix-free, depends at time $\mathsf{t}$ only on the previous transformed outputs, and the decoder reconstructs the quantizer output exactly.

Time-average code-length conjecture. The time-average expected codeword lengths should satisfy

$$\lim_{\mathsf{T}\rightarrow\infty}\frac{1}{\mathsf{T}}\sum_{\mathsf{i}=0}^{\mathsf{T}-1}\mathbb{E}[\ell({\boldsymbol{\mathbf{MakeLowercase{a}}}}_{\mathsf{t}})]=H({\boldsymbol{\mathbf{MakeLowercase{q}}}})+2.$$

This would extend stationary-source coding guarantees to the asymptotically stationary quantizer outputs while preserving the LQG control-performance constraint. The source explicitly says that proving or disproving the conjecture is future work, so it remains open.