Hausdorff dimension of uniformly non flat sets with topology

Let $d$ be an integer, and let $E$ be a nonempty closed subset of $\mathbb{R}^n$. Assume that $E$ is locally uniformly non flat, in the sense that for $x\in E$ and $r > 0$ small, $E\cap B(x,r)$ never stays $\varepsilon_0 r$-close to an affine $d$-plane. Also suppose that $E$ satisfies locally uniformly some appropriate $d$-dimensional topological nondegeneracy condition, like Semmes' Condition B. Then the Hausdorff dimension of $E$ is strictly larger than $d$. We see this as an application of uniform rectifiability results on Almgren quasiminimal (restricted) sets.