Graphs that are not minimal for conformal dimension

We construct functions $f\text{\hspace{0.17em}⁣}:[0,1]\to [0,1]$ whose graph as a subset of ${\mathbb{R}}^2$ has Hausdorff dimension greater than any given value $\alpha \in (1,2)$ but conformal dimension $1$. These functions have the property that a positive proportion of level sets have positive codimension-$1$ measure. This result gives a negative answer to a question of Binder–Hakobyan–Li. We also give a function whose graph has Hausdorff dimension $2$ but conformal dimension $1$. The construction is based on the author’s previous solution to the inverse absolute continuity problem for quasisymmetric mappings.