The multiplicative-noise extension conjecture for uniform dimension theorems

Let $u$ be the random field associated with a parabolic stochastic partial differential equation with multiplicative noise, and let $\sigma(v)$ be its $p\times p$ noise coefficient matrix. Define

$$\lambda(v)=\inf_{x\in\mathbb{R}^p\setminus\{0\}}\frac{\|\sigma(v)x\|^2}{\|x\|^2}.$$

A set $A\subset\mathbb{R}^p$ is polar for $u$ if $u$ almost surely does not hit $A$.

Multiplicative-noise extension conjecture. Theorem (ii) and (iii) for the additive-noise equation remain valid in the presence of multiplicative noise, provided that

$$\lambda^{-1}\{0\}$$

is polar for $u$.

This is posed as a likely extension of the additive-noise uniform dimension results despite the nonlocal nature of uniform dimension statements. The source says that the authors have no idea how to prove it and provides no evidence of a resolution.