On volumes of simplices in intermediate dimensions

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_