Zero-freeness in a tube for the Sherrington–Kirkpatrick partition function

Let $Z_{\bm{G}}(\beta)$ denote the partition function of the Sherrington–Kirkpatrick model, let $\varepsilon>0$, and let $\beta_{\mathsf{2nd}}$ be the second-moment threshold. A constant-width tube is a tube $\mathcal{T}$ in the complex inverse-temperature plane containing the real interval $[0,(1-\varepsilon)\cdot\beta_{\mathsf{2nd}}]$. Zero-freeness in a tube. There exists a constant-width tube $\mathcal{T}\supseteq [0,(1-\varepsilon)\cdot\beta_{\mathsf{2nd}}]$ such that $Z_{\bm{G}}(\beta)$ is zero-free in $\mathcal{T}$ with high probability. This conjecture would strengthen the paper’s algorithmic results by replacing the existence of a large subset of successful inverse temperatures with a contiguous complex zero-free region. The supplied text does not indicate whether the conjecture has been resolved.