The Davenport constant of an interval: a proof that $\mathsf{D}=\chi$

For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\text{\hspace{0.17em}⁣}[-m,M]\text{\hspace{0.17em}⁣}]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\text{\hspace{0.17em}⁣}[-m,M]\text{\hspace{0.17em}⁣}]$. We prove the conjecture that it is equal to $m+M-r$ where $r$ is the smallest integer which can be decomposed as a sum of two non-negative integers $t_1$ and $t_2$ ($r=t_1+t_2$) having the property that $\gcd (M-t_1,m-t_2)=1$.