Asymptotic classification and uniqueness of one-sided minimisers

Assume the hypotheses of the one-force-one-solution conjecture, and work almost surely. A one-sided minimiser is a curve defined on $(-\infty,t]$ such that compact perturbations fixing its endpoint and infinite tail do not decrease the action. One-sided minimiser conjecture. Every one-sided minimiser $\gamma$ has an asymptotic slope

$$a=\lim_{s\to-\infty}\frac{\gamma(s)}{s}\in\mathbb{R}^d.$$

For every fixed $a\in\mathbb{R}^d$ and every endpoint $(t,x)$ there is a minimiser with slope $a$; for Lebesgue-almost-every endpoint it is unique; minimisers with the same slope satisfy

$$\lim_{s\to-\infty}|\gamma_1(s)-\gamma_2(s)|=0;$$

for a minimiser $\gamma_{t,x}$ with slope $a$, the function defined by the stated action difference is a global Hamilton–Jacobi solution $\Phi^a$, equal to $\Phi_b$ for a uniquely defined $b=b(a)$; and $\Phi^a$ is continuous and locally Lipschitz, with nonuniqueness points exactly the shocks described in the claim. These assertions are the inviscid counterpart of one-force-one-solution and connect global solutions with one-sided minimisers; the source provides no resolution.