Differentiability and weighted Laplacian formula on RCD spaces

Let $(X,\mathsf{d},\mathfrak{m})$ be the RCD space under discussion, let $k=\dim(X,\mathsf{d},\mathfrak{m})$, and let $\mathcal{R}_k^*$ be the full-measure regular set. Let $\phi$ be the function defined by equation (9), and for $f\in D(\Delta)$ let $\mathrm{Hess}_f$ be its Hessian. Differentiability and weighted Laplacian conjecture. The function $\phi$ is differentiable for $\mathfrak{m}$-almost every $x\in\mathcal{R}_k^*$. Moreover, for every $f\in D(\Delta)$,

$$\Delta f(x)=\operatorname{tr}(\mathrm{Hess}_f)(x)+\langle\nabla\log\phi,\nabla f\rangle(x)$$

holds for $\mathfrak{m}$-almost every $x\in X$. This is the paper's final conjecture, giving a pointwise weighted correction to the Laplacian–Hessian trace identity; the supplied context gives no resolution.