Stochastic codimension conjecture for Gaussian-process images

Let $X$ be the Gaussian process from the almost-sure image-dimension zero-one law, let $E\subset[0,1]$ be Borel, and let $\dim_{\delta}(E)$ and $\overline{\dim}_{\delta,M}(E)$ denote the lower and upper metric-Minkowski dimensions associated with the canonical metric $\delta$. Write $\mathbf{c}(E)$ for the almost-sure value of $\dim_{\operatorname{euc}}X(E)$, and let $d$ be the Euclidean dimension of the state space. Assume

$$\dim_{\delta}(E)=\overline{\dim}_{\delta,M}(E)\leq \mathbf{c}(E)\leq d.$$

Stochastic codimension conjecture. For every Borel set $F$ in the state space,

$$\mathbb{P}\{X(E)\cap F\neq\varnothing\}>0 \quad\text{if }\dim_{\operatorname{euc}}(F)>d-\mathbf{c}(E),$$

and

$$\mathbb{P}\{X(E)\cap F\neq\varnothing\}=0 \quad\text{if }\dim_{\operatorname{euc}}(F)<d-\mathbf{c}(E).$$

Equivalently, the conjectured stochastic codimension is

$$\operatorname{codim}(X(E))=d-\mathbf{c}(E).$$

The claim would identify the hitting-probability threshold of the image with its almost-sure Euclidean Hausdorff dimension, extending the known estimates beyond the regularity regime of Condition $\mathbf{(C_{0+})}$. The source states that the value of $\mathbf{c}(E)$ is unknown for highly irregular processes, so this remains open.