The point-count upper bound conjecture for nonsingular hypersurfaces in even-dimensional projective spaces

Let $m\geq 4$ be an even integer, and let $\mathcal{X}$ be a nonsingular hypersurface of degree $d$ in $\mathbb{P}^m$ over $\mathbb{F}_q$. Write $N_q(\mathcal{X})$ for the number of $\mathbb{F}_q$-rational points of $\mathcal{X}$, and let $\theta_q(r)$ denote the number of points of $\mathbb{P}^r$ over $\mathbb{F}_q$. The point-count upper bound conjecture. One might have

$$N_q(\mathcal{X})\leq \theta_q\left(\frac{m}{2}-1\right)\left((d-1)q^{\frac{m}{2}}+1\right).$$

The preceding argument shows that the general upper bound obtained for even-dimensional projective spaces is not attained, but it does not establish this sharper bound. Thus the displayed inequality is proposed as a conjectural bound for nonsingular hypersurfaces when $m$ is even.