Conjecture on improving the weak-noise threshold for Hölder regularity

Fix $\theta\in\mathbb{R}$, and let $u_\theta$ be the invariant measure associated with the solution $V^{(\theta)}$ of the SPDE started from the constant initial profile $\theta$. Assume the Hölder regularity condition, and write the weak-noise condition with threshold $(1-\beta)/(4d)$, where $\beta$ is the constant in. Weak-noise threshold conjecture. The conclusion of Theorem, namely that $u_\theta$ is supported on $C_{\mathrm{loc}}^\varepsilon(\mathbb{R}^d)$ for some $\varepsilon>0$, continues to hold when the threshold $(1-\beta)/(4d)$ is replaced by a larger quantity of the form $c(\beta)/d$. Equivalently, the asymptotically optimal Burkholder–Davis–Gundy constant does not give the sharp result in this setting. The paper states this as an unresolved problem; the improved threshold and sharpness question remain open.