Exceptional-plane dimensions for the three-dimensional Bargmann–Fock field

Let $F$ be a three-dimensional Bargmann–Fock field on $\mathbb{R}^3$. For each $t\in\mathbb{R}$, let $f^{\mathrm{hor}}(t)$ be the two-dimensional Bargmann–Fock field obtained by restricting $F$ to the horizontal plane $\{(x,y,t):x,y\in\mathbb{R}\}$. Exceptional-plane conjecture. Almost surely, the Hausdorff dimension of the set of times $t$ for which $\{f^{\mathrm{hor}}(t)>0\}$ contains an unbounded component is

$$\frac{31}{36},$$

and the Hausdorff dimension of the set of times $t$ for which $\{f^{\mathrm{hor}}(t)=0\}$ contains an unbounded component is

$$\frac{2}{3}.$$

This predicts the dimensions of exceptional horizontal planes in three-dimensional Bargmann–Fock percolation. The claim is presented as a consequence that would follow from the preceding exceptional-time conjecture; no proof or resolution is given in the source.