Edoukou's point-count conjecture for projective algebraic sets

Let $X\subset\mathbb{P}^{n}(\mathbb{F}_q)$ be a projective algebraic set of degree $d$ and dimension $s$. Writing $\pi_j=|\mathbb{P}^j(\mathbb{F}_q)|$, Edoukou's point-count conjecture.

$$|X(\mathbb{F}_q)|\le dq^s+\pi_{s-1}.$$

The source remarks that this is true in codimension one, where it is the Tsfasman–Serre–Sørensen upper bound for hypersurfaces, and that the bound for intersections of two quadrics with no common hyperplane is a special case. The general assertion is presented as a conjecture.