Bagchi–Datta generalized lower bound conjecture for triangulated manifolds

Let $\Delta$ be a connected triangulated $(d-1)$-manifold without boundary, and let $h_r(\Delta)$ denote its $h$-numbers and $\beta_{j-1}(\Delta)$ its Betti numbers. Bagchi–Datta's generalized lower bound conjecture for triangulated manifolds.

(i) For $r=1,2,\ldots,\left\lfloor\frac d2\right\rfloor$,

$$h_r(\Delta)\geq h_{r-1}(\Delta)+\binom{d+1}{r}\sum_{j=1}^r(-1)^{r-j}\beta_{j-1}(\Delta).$$

(ii) If equality holds for some $r<\frac d2$ in (i), then $\Delta$ is locally $(r-1)$-stacked. This conjecture extends the generalized lower bound conjecture from polytopal spheres to triangulated manifolds; the source does not state a resolution.