Equitable full-cookie distribution conjecture

Let $r$, $n$, and $m$ be positive integers. A group of $r$ children has $n$ cookies, each with some amount of one or more of $m$ possible kinds of frosting.

Equitable distribution conjecture. It should be possible to cut at most $(r-1)m$ cookies and distribute them so that each child has the same amount of each kind of frosting and at least

$$\left\lfloor \frac{n-m(r-1)}{r}\right\rfloor$$

full cookies.

The bound would be optimal because an equitable distribution of all full cookies can give each child at most $\left\lfloor\frac{n-m(r-1)}{r}\right\rfloor$ full cookies. The conjecture strengthens the paper's theorem by improving its guaranteed number of full cookies while retaining the optimal cut bound.