Entropy representation by prefix Kolmogorov complexity

Let $\mu$ be the universal lower-semicomputable semi-density matrix from the preceding conjecture. Let $\ket{1},\ket{2},\ldots$ be a computable orthogonal sequence of states, and define real-valued functions by

$$\overline{H}(\ket{\psi})=-\braket{\psi|(\log\mu)\psi},\qquad \underline{H}(\ket{\psi})=-\log\braket{\psi|\mu\psi}.$$

Here $K(i)$ denotes prefix Kolmogorov complexity.

Entropy-complexity conjecture. For $H=\overline{H}$ or $H=\underline{H}$, one has

$$H(\ket{i})=K(i)+O(1).$$

This is intended as a quantum analogue of the relationship between classical algorithmic entropy and prefix complexity. The supplied text gives no resolution status for this claim.