Before revision:  To complement result (ii), we show that the mim-width of {\mathcal H}-convex graphs is unbounded if {\mathcal H} is the set of trees with arbitrarily large maximum degree or  arbitrarily large number of vertices of degree at least 3.
After revision:  To complement result (ii), we show that the mim-width of {\mathcal H}-convex graphs is unbounded if {\mathcal H} is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.