The uniform dimension conjecture for parabolic SPDEs on the torus

Let $u$ be the random field and let $\sigma(v)$ be the coefficient matrix defining the multiplicative noise, with $v a0 in \mathbb{R}^p$. A set $A\subset\mathbb{R}^p$ is polar for $u$ if $u$ almost surely does not hit $A$. For a compact set $F\subset\mathbb{T}$ and $t>0$, write $u(\{t\}\times F)$ for the corresponding spatial image, and let $\operatorname{dim_H}$ denote Hausdorff dimension.

Uniform dimension conjecture. The uniform dimension theorem stated for $p\geqslant4$ is valid whenever $p\geqslant2$, provided that

$$\{v\in\mathbb{R}^p:\inf_{\|x\|=1}\|\sigma(v)x\|=0\}$$

is polar for $u$.

The conjecture would extend the uniform Hausdorff-dimension formula simultaneously to all compact spatial sets and positive times under the polarity condition. The source gives no resolution and presents it as a conjecture motivated by the known theorem and by the corresponding Brownian-motion dimension result.