Conjecture on the Hausdorff dimensions of KPZ fixed-point non-uniqueness times

Let $h_0$ be an initial condition such that the KPZ fixed point $\mathfrak{h}=\mathfrak{h}(\cdot;h_0)$ attains a maximum at every time $t>0$ almost surely. Define

$$\operatorname*{\arg\,\max}\mathfrak{h}_t=\{y\in\mathbb{R}:\max_{x\in\mathbb{R}}\mathfrak{h}_t(x)=\mathfrak{h}_t(y)\}.$$

For $k\in\mathbb{N}$, define the non-uniqueness times

$$\mathcal{T}_k(h_0)=\{t\in(0,\infty):|\operatorname*{\arg\,\max}\mathfrak{h}_t|=k\},\qquad \mathcal{T}_{\geq k}(h_0)=\{t\in(0,\infty):|\operatorname*{\arg\,\max}\mathfrak{h}_t|\geq k\}.$$

Here $\dim$ denotes Hausdorff dimension.

Non-uniqueness-times conjecture. For nice enough initial conditions $h_0$, almost surely $\mathcal{T}_{\geq2}(h_0)$ is dense in $[0,\infty)$, $\mathcal{T}_{\geq5}(h_0)$ is empty, and

$$\dim\mathcal{T}_2(h_0)=\frac{2}{3},\qquad \dim\mathcal{T}_3(h_0)=\frac{1}{3},\qquad \dim\mathcal{T}_4(h_0)=0.$$

Earlier work established Hausdorff dimension $2/3$ for the set of times with at least two maximizers under suitable assumptions, including the narrow-wedge case. The conjecture refines that result by predicting density, the impossibility of five or more maximizers, and the dimensions of the exact multiplicity strata.