Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, \partial x_j} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}$$ or $$\Delta_\infty u(x) = \frac{\langle Du, D^2 u \, Du \rangle}{|Du|^2} = \frac{1}{|Du|^2} \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, \partial x_j} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j};$$ the set of infinity harmonic functions remains unchanged with either formulation. The natural space for solutions is $W^{1,\infty}$.

It is known that solutions are differentiable everywhere, but continuous differentiability is open.

I am interested in the integrability of the second order derivative $D^2 u$, or even with integrability of the Laplacian $\Delta u$, of infinity harmonic functions.

In particular, is it known if $D^2 u$ is a signed Radon measure, or, even better, do we have $u \in W^{2,1}$?

These are true for the Aronsson solution $$u(x,y) = x^{4/3}-y^{4/3},$$ which is usually taken as the model case for infinity harmonic functions. Is there any known technique for proving integrability results for $\infty$-harmonic functions?