Conjecture for the Mattila-Sjölin problem for k-distance sets

Let $\mathbb{F}_q$ be a finite field, let $E\subset \mathbb{F}_q^d$ with $d\geq 2$, and for $x=(x_1,\ldots,x_d)\in\mathbb{F}_q^d$ let $\mathcal{Z}(x)$ be the number of zero coordinates of $x$. For a fixed integer $k\in\{1,2,\ldots,d\}$, define

$$\lVert x\rVert_k= \begin{cases} \lVert x\rVert & \text{if }0\leq\mathcal{Z}(x)\leq k-1,\\ 0 & \text{otherwise,} \end{cases}$$

and define the $k$-distance set by

$$D_k(E)=\{\lVert x-y\rVert_k:x,y\in E\}.$$

The conjecture for the Mattila-Sjölin problem for $k$-distance sets. Assume that $C$ is a sufficiently large constant independent of $q$. Then, for every integer $k=1,2,\ldots,d$, if $d$ is odd and

$$|E|\geq Cq^{\max\{(d+1)/2,\,d-k\}},$$

then $D_k(E)=\mathbb{F}_q$, while if $d$ is even and

$$|E|\geq Cq^{\max\{d/2,\,d-k\}},$$

then $D_k(E)=\mathbb{F}_q$.

The claim strengthens the finite-field Mattila-Sjölin problem because $D_k(E)\subset\Delta(E)$, and the example $E=\mathbb{F}_q^{d-k}\times\{\mathbf{0}\}$ shows that the threshold cannot be below $d-k$. The source presents this as a proposed conjecture; its resolution is not established in the supplied text.