Aubry et al.'s point-count conjecture for weighted hypersurfaces

Let $k=\mathbb{F}_q$ be a finite field, let $\mathbb{P}_{\mathbf{w}}^n$ be weighted projective space with weights $\mathbf{w}=(w_0,w_1,\ldots,w_n)$, and write

$$p_n=q^n+\cdots+q+1.$$

For a weighted homogeneous polynomial $F$ of degree $d$, define

$$e_q(d;w_0,w_1,\ldots,w_n)=\max\{|X(\mathbb{F}_q)|:X=V(F)\}.$$

Aubry et al.'s point-count conjecture. If $1=w_0\leq w_1\leq w_2\leq\cdots\leq w_n$ and $\operatorname{lcm}(w_1,w_2,\ldots,w_n)\mid d$, then

$$e_q(d;w_0,w_1,\ldots,w_n)=\min\left\{p_n,\frac{d}{w_1}q^{n-1}+p_{n-2}\right\}.$$

It is enough to prove the assertion for $d\leq w_1(q+1)$, and it is known for weighted projective spaces of dimensions one and two in the cases stated in the source.