Hodge–Betti convergence conjecture for non-collapsed Ricci limit spaces

Let

$$M_i^n\stackrel{\mathrm{GH}}{\to}X^n$$

be a non-collapsed sequence of closed Riemannian manifolds with $\mathrm{Ric}\ge -(n-1)$, converging in the Gromov–Hausdorff sense to a compact Ricci limit space $X^n$. Let $b_{H,1}(X^n)$ denote the dimension of the space of harmonic $1$-forms on $X^n$ in Gigli's sense. Hodge–Betti convergence conjecture. Then, for all sufficiently large $i$,

$$b_1(M_i^n)=b_{H,1}(X^n),$$

and

$$b_{H,1}(X^n)=b_1(X^n).$$

The conjecture asks whether first Betti numbers and harmonic first-form dimensions behave continuously under these non-collapsed Ricci-limit convergences; the supplied text gives no resolution.