For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
How big can $\mathsf{LCM}(\mathcal R(a,b))$ be if $a,b\in\big[\frac r2,r\big]$ hold and are coprime?