Conjectured construction of solutions to the constrained Hamilton-Jacobi problem

Restrict to dimension $n=1$. Let $\tilde u$ solve the unconstrained Hamilton-Jacobi problem associated with the linearized kinetic equation, and let $\phi_r$ be its fundamental solution. For initial data $u_0$, define

$$\tilde u(t,x,v)=\min_{(y,w)}\left(\phi_r(t,x-y,v;w)+u_0(y,w)\right).$$

The constrained problem additionally requires $\min_v u(t,x,v)\geq 0$.

Constrained-problem construction conjecture. First solve the unconstrained problem using the formula above. Then, whenever $\min \tilde u\leq 0$, replace $\tilde u$ by $v^2/2$ on the zone $v<x/t$ for $x>0$, respectively on the zone $v>x/t$ for $x<0$. This procedure should determine the minimal value $\min_v u$ for every $(t,x)$, attained necessarily at $v=0$.

The proposed truncation would provide a solution of the constrained viscosity problem, but the authors state that they were unable to prove its validity. It is presented as a procedure analogous to the approach of Evans and Souganidis.