Density conjecture for non-uniqueness times in the KPZ fixed point

Let $\mathfrak{h}_0$ satisfy Assumption~, and let $\mathcal{T}_\infty(\mathfrak{h})$ be the set of times at which the KPZ fixed point has non-unique maximizers. Density conjecture. For every such initial condition, $\mathcal{T}_\infty(\mathfrak{h})$ is almost surely dense in $[0,\infty)$. For narrow-wedge initial data, the paper proves nonemptiness of the corresponding exceptional-time set at every prescribed time horizon; the conjecture extends this phenomenon to general initial data and strengthens it to almost-sure density.