Divisibility characterization for centred Catalan sets

Let $S$ be a centred Catalan set, let $i$ be an integer, and define the rising factorial by

$$(x)_k:=x(x+1)\cdots(x+k-1).$$

Divisibility conjecture. One has

$$\{-i,\ldots,i\}\subseteq S\quad\Longleftrightarrow\quad \prod_{k=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}(l+1+3k)_{i-2k}\textnormal{ divides }w_l(S).$$

This conjecturally relates the central interval contained in a centred Catalan set to explicit factors of its weight function. The supplied text gives no resolution or additional hypotheses on $i$.