Nonnegativity conjecture for the second gamma-number of discrete pseudomanifolds

Let $M$ be a discrete $d$-pseudomanifold. Define its second gamma-number, using the numbers $f_0(M)$ and $f_1(M)$ of vertices and edges, by

$$\gamma_2(M)=f_1(M)-(2d-1)f_0(M)+(d-1)(2d+2).$$

Nonnegativity conjecture for discrete pseudomanifolds.

$$\gamma_2(M)\geq 0.$$

The paper verifies this inequality for every discrete $d$-pseudomanifold with at most $2d+7$ vertices, based on its classifications, but gives no resolution for arbitrary discrete pseudomanifolds.