Nonexistence of convex solutions to HRMA beyond the convex lifespan

Determine whether there exists no convex (i.e., Lipschitz) weak solution beyond the convex lifespan T_cvx for the Cauchy problem of the homogeneous real Monge–Ampère equation det D^2_{(s,x)} φ = 0 on [0,T] × R^n with initial data φ(0,x) = v_0(x) and ∂_s φ(0,x) = v_0(x), where v_0 is smooth and strictly convex with ∇v_0(R^n) equal to a fixed Delzant polytope P, under the branch condition that φ is convex in (s,x) and ∇_x φ(s,·)(R^n) = P for each s.

Background

In the toric Kähler setting, geodesics in the space of Kähler metrics reduce to solutions of the homogeneous real Monge–Ampère equation (HRMA). For smooth, strictly convex initial data v_0 with ∇v_0(R^n) equal to a Delzant polytope P, the natural candidate solution solves HRMA up to a finite time T_cvx (the convex lifespan) and ceases to solve it precisely when the associated Legendre-dual function loses convexity.

Rubinstein–Zelditch proved there is no C^1 continuation solving HRMA beyond T_cvx by relating the problem to a Hamilton–Jacobi equation. The remaining question is whether even a non-smooth, merely convex (hence Lipschitz) weak solution can exist past T_cvx.