Finer-graded Lefschetz inequality conjecture for free cyclic actions

Let $\Delta$ be an orientable $\mathbb{k}$-homology manifold of dimension $d-1$ admitting a free action by the cyclic group $\mathbb{Z}/p\mathbb{Z}$. For the finer $({\mathbb Z}\times \mathbb{Z}/p\mathbb{Z})$-grading on $\mathbb{k}[\Delta]/\Sigma(\Theta;\mathbb{k}[\Delta])$, write the graded pieces as $\left(\mathbb{k}[\Delta]/\Sigma(\Theta;\mathbb{k}[\Delta])\right)_i^j$. Finer-graded Lefschetz inequality conjecture. There exists $m$ such that

$$\dim_\mathbb{k} \left(\mathbb{k}[\Delta]/\Sigma(\Theta;\mathbb{k}[\Delta])\right)_{i-1}^j\le \dim_\mathbb{k} \left(\mathbb{k}[\Delta]/\Sigma(\Theta;\mathbb{k}[\Delta])\right)_i^{j+m}$$

for $1\le i\le \lfloor d/2\rfloor$ and $0\le j\le p-1$. This refines the usual Lefschetz-type inequalities by requiring them in the finer grading; the general assertion is presented as a conjecture, with only partial evidence from known special cases and Lefschetz elements.