Conjecture on zeroes of functions in the evaluation space

Let $V$ be the function space under consideration, and let $\mathcal{B}$ be the evaluation set. Let $l$, $q$, $k$, and $i$ be the parameters used in the construction, and let the code's lower bound for the distance be the one given in Theorem.

Zero-count conjecture. Any function $f\in V$ will have fewer than

$$lq^{k}+(k-1)(q-1)q^{k-1}+(q-2)q^{k-1}$$

zeroes in $\mathcal{B}$, and therefore the lower bound for the distance in Theorem will not be attained for $i\geq 3$.

This conjecture predicts that the previously obtained distance lower bound is not sharp in the stated range, by ruling out functions attaining the corresponding maximum number of zeroes on the evaluation set. Its status is not resolved in the supplied text.