Sharpness conjecture for the Serre-type lower bound in weighted projective spaces

Let $\mathbb{P}(a_0,a_1,\ldots,a_m)$ be a weighted projective space over $\mathbb{F}_q$, let $e_q(d;1,a_1,a_2,\ldots,a_m)$ be the minimum number of $\mathbb{F}_q$-rational zeros of a nonzero weighted-homogeneous polynomial of degree $d$, and suppose that $a_0=1$ and $\operatorname{lcm}(a_1,a_2,\ldots,a_m)\mid d$. Order the weights so that $a_1\leq a_2\leq\ldots\leq a_m$. Sharpness conjecture. The lower bound from the preceding lemma is sharp, namely

$$e_q(d;1,a_1,a_2,\ldots,a_m)=\min\left\{p_m,\frac{d}{a_1}q^{m-1}+p_{m-2}\right\}.$$

The conjecture predicts the exact minimum number of rational zeros in this weighted-projective setting under the stated divisibility and weight-ordering assumptions.