Injection dimension of weighted projective spaces with two coprime weights

Let $d,e\geq 2$ be relatively prime integers and let $r,s\geq 2$. Let $\mathbb{P}(\underbrace{d,\ldots,d}_{r},\underbrace{e,\ldots,e}_{s})$ denote the weighted projective space with $r$ coordinates of weight $d$ and $s$ coordinates of weight $e$, and let $\gamma$ denote its injection dimension. The weighted-projective-space conjecture. One expects

$$\gamma\mathopen{}\mathclose\bgroup\originalleft(\mathbb{P}(\underbrace{d,\ldots,d}_{r},\underbrace{e,\ldots,e}_{s})\aftergroup\egroup\originalright)=2(r+s-1)-1.$$

This predicts the exact injection dimension in a family where the preceding general lower bound is not always sharp; the source provides no proof or resolution of the expectation.