Symmetry inequality conjecture for the entropy-difference penalty function

Let $d\geq 2$ and let $M(\Delta,d)$ denote the penalty function defined by the minimization over pairs of distributions with entropy difference $\Delta$. For $\Delta\in[0,\log d]$, symmetry inequality conjecture.

$$M(\Delta,d)\geq M(-\Delta,d).$$

Numerical plots for many values of $d$ support the inequality, and it is consistent with the analytical results established earlier in the paper, but no proof is known. If true, it would imply the stronger lower bound $R^*\geq M(-(S_{\max}-S_{\min})/2,d)$ for the universal-coding redundancy.