Measure-valued Hessian conjecture for Lipschitz subsolutions

Let $w$ be a Lipschitz subsolution to the generalized $k$-convexity equation for $\beta=1$. A measure-valued Hessian conjecture asserts that there exist signed Borel measures $mu^{ij}=mu^{ji}$ such that

$$\int_{\mathbb{R}^n} w \, \partial_{ij} \phi(x) \, dx=\int_{\mathbb{R}^n} \phi(x) \, d\mu^{ij}(x),\qquad i,j=1,2,\ldots,n,$$

for every $\phi\in C_c^\infty(\mathbb{R}^n)$. This would say that every such subsolution has a distributional Hessian represented componentwise by signed Borel measures. The statement concerns second-order regularity of viscosity subsolutions, but the supplied text does not establish whether it is proved or remains open.