Hausdorff-dimension conjecture for exceptional triples in the directed landscape

Let $X$ be the set of triples $(y,s,t)$ with $y\in\mathbb{R}$ and $s<t$. Equip $X$ with the metric

$$d((y_1,s_1,t_1),(y_2,s_2,t_2))=|t_1-t_2|+|s_1-s_2|+|y_1-y_2|^{3/2}.$$

For $k\geq2$, let $E_k$ be the subset of triples with $|s-t|\geq1$ for which there are at least $k$ maximizers $x_1,\ldots,x_k$ of $x\mapsto\mathcal{L}(y,s;x,t)$ satisfying $|x_i-x_j|\geq1$ for all $1\leq i<j\leq k$. Directed-landscape exceptional-set dimension conjecture. Almost surely,

$$\dim_H(E_k)=\max\{3-k/3,0\},$$

and $E_{10}$ is empty. The metric is adapted to KPZ space-time scaling, making the Hausdorff-dimension assertion a natural space-time analogue of exceptional-time dimension questions; these claims are presented as conjectural in the paper.