Symmetric $q$-stability conjecture

For $q\geq 1$, let $\ln_q:(0,+\infty)\to\mathbb{R}$ be the $q$-logarithm

$$\ln_q(t):=\begin{cases}\frac{t^{q-1}-1}{q-1}, & q\neq 1,\ln(t), & q=1,\end{cases}$$

and define $\Phi_q^{\mathrm{sym}}(t)=t\ln_q(t)+(1-t)\ln_q(1-t)$ and the associated symmetric $q$-stability by $\mathbf{Stab}_q^{\mathrm{sym}}$. A Boolean function $f_d$ is a dictator function if $f_d=1\{x_k=1\}$ or $1\{x_k=-1\}$ for some $1\leq k\leq n$; a Boolean function is balanced when $\mathbb{E}f=1/2$. Symmetric $q$-Stability Conjecture. For $0\leq\rho\leq 1$ and $q\in[1,9]$, it holds that

$$\mathbf{Stab}_q^{\mathrm{sym}}(f)\leq\mathbf{Stab}_q^{\mathrm{sym}}(f_d)$$

for all balanced Boolean functions $f$, where $f_d$ is a dictator function. This conjecture asserts that dictators maximize symmetric $q$-stability among balanced Boolean functions; its status is not resolved in the supplied source context.