Uniqueness of free-time controlled transport in Eulerian form

Let $\mu, \nu$ be compactly supported probability measures on $\mathbb{R}^n$ with disjoint supports, at least one of them absolutely continuous. Let $A \subset \mathbb{R}^m$ be closed, let $K(t, q, a) \ge 0$ be continuous, and let $f(q, a)$ be Lipschitz in $q$, continuous in $a$, with $\{f(q, a) : a \in A\}$ spanning $\mathbb{R}^n$ at each $q$. Require $$\sup_q |f(q, a)| < \infty, \quad |f(q, a)| \le \Gamma_0(1 + |a|), \quad \liminf_{|a| \to \infty} \frac{K(t, q, a)}{|a|} = \infty, \quad \liminf_{T \to \infty} \int_0^T \inf_{q, a} K(t, q, a)\, dt = \infty.$$ Assume that $H(t, q, p) = \sup_a \{p \cdot f(q, a) - K(t, q, a)\}$ is smooth with $H \sim |p|^\beta$, $1 < \beta < \infty$, uniformly in $(t, q)$, that its maximizing control is unique and continuous in $(t, q, p)$, and that $K$ is either strictly increasing or strictly decreasing in $t$.

Minimize $\int_0^\infty \int K\, d\rho_t\, dt$ over weak*-measurable families $\rho_t$ of nonnegative finite measures on $\mathbb{R}^n \times A$ and nonnegative finite stopping measures $\sigma$ on $[0, \infty) \times \mathbb{R}^n$ with spatial marginal $\nu$, subject to $$\int w\, d\sigma = \int w(0, x)\, d\mu(x) + \int_0^\infty \int (\partial_t w + f \cdot \nabla_q w)\, d\rho_t\, dt$$ for test functions $w \in C^1(\mathbb{R} \times \mathbb{R}^n)$, with integrals understood. Ghoussoub, Kim and Palmer proved that a minimizer exists, that the optimal transport plan for the cost of steering $x$ to $y$ at a free end time is unique, and that embedding it gives a minimizer. They note that the minimizer is unique if the optimal dual value function is $C^1$.

Problem. Is the minimizing pair $(\rho_t\, dt, \sigma)$ unique? The occupation measure $\rho_t\, dt$ includes the controls, and changes of $\rho_t$ on null sets of times do not count. The coercivity above is not required to be uniform in $(t, q)$.