Dimension bounds for relative character varieties on the projective line with three punctures $G=GL(r), O(r), Sp(r)$

We consider relative character varieties on $\mathbb{P}^1\backslash\{0,1,\infty\}$ with $G=GL(r), O(r)$, or $Sp(r)$. Using a diagrammatic method of Simpson's, we give an explicit linear upper bound $R(d)$ on the rank $r$ of an MC-minimal character variety of dimension $d>2$. An arbitrary character variety is isomorphic, via Katz's middle convolution, to one satisfying the bound. For the general linear and non-overlapping quadratic cases, the bounds we give are the sharpest possible using this method.