Gigli's Laplacian–Hessian trace characterization conjecture

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space. For $f\in D(\Delta)$, let $\mathrm{Hess}_f$ be its Hessian, viewed as a $(0,2)$-type $L^2$ tensor, and let $\operatorname{tr}(\mathrm{Hess}_f)$ denote its trace. Gigli's conjecture. The following conditions are equivalent: (a) for every $f\in D(\Delta)$,

$$\Delta f(x)=\operatorname{tr}(\mathrm{Hess}_f)(x)$$

for $\mathfrak{m}$-almost every $x\in X$; (b) $\mathfrak{m}=b\mathcal{H}^k$ for some $b,k\in(0,\infty)$. The source proposes this equivalence by combining a result of Brue--Semola with the preceding conjectures; it gives no resolution.