Héra's conjecture on the dimension of unions of k-planes

Let $0<k\le d<n$ be integers and let $\beta\in[0,k+1]$. Let $\mathcal{V}\subset A(k,n)$ be a set of $k$-planes in $\mathbb{R}^n$, with

$$\dim(\mathcal{V})=(k+1)(d-k)+\beta.$$

Here, $A(k,n)$ denotes the affine Grassmannian in $\mathbb{R}^n$, and $\dim(\mathcal{V})$ denotes the Hausdorff dimension of $\mathcal{V}$ as a subset of the Riemannian manifold $A(k,n)$. Héra's conjecture.

$$\dim\left(\bigcup_{V\in\mathcal{V}}V\right)\ge d+\min\{1,\beta\}.$$

The paper proves this conjecture, giving a sharp lower bound for the Hausdorff dimension of unions of families of affine $k$-planes.