Mirror and phase-flip symmetry conjecture for real minimal qubit PI codes

Let a minimal qubit permutation-invariant code have real codeword coefficients, written as $\alpha_i$ and $\beta_i$, with minimal block length $n_{\rm \min}$ for error weight $t$. The observed relation is

$$\beta_i=(-1)^i\alpha_{n_{\rm \min}-i}.$$

Here the absolute-value relation is called mirror symmetry and the sign relation is called phase-flip symmetry. Real minimal PI-code symmetry conjecture. All minimal qubit permutation-invariant codes with real coefficients exhibit both the mirror and phase-flip symmetries in their codewords. The conjecture concerns the structure of every real minimal code beyond the numerically observed examples.