Lower bound conjecture for connected non-simply connected triangulated manifolds

Let $d\geq 3$, and let $X$ be a connected, non-simply connected triangulated $d$-manifold. Lower bound conjecture for non-simply connected manifolds. Its face numbers satisfy

$$f_j(X)\geq\begin{cases} \binom{d+1}{j}f_0(X),&1\leq j<d,\\ df_0(X),&j=d. \end{cases}$$

Equality holds for some $j$ if and only if $X$ is obtained from a stacked $d$-sphere by an elementary handle addition. This extends lower-bound phenomena from spheres to connected non-simply connected triangulated manifolds. The source presents it as a conjecture and gives no resolution; related lower-bound inequalities were known in more restricted settings.