Turán bound conjecture for higher-dimensional grids

For positive integers $d$ and $t$, let $F_t^{(d)}$ be the $d$-dimensional grid with vertex set $[t]^d$, where two vertices are adjacent when they differ by exactly one in exactly one coordinate. For a positive integer $n$, let $\mathrm{ex}(n,F_t^{(d)})$ denote the maximum number of edges in an $F_t^{(d)}$-free graph on $n$ vertices. Higher-dimensional grid conjecture. There is a constant $C$ such that

$$\mathrm{ex}(n,F_t^{(d)})\leq Cn^{2-1/d}.$$

This is suggested by the fact that $F_t^{(d)}$ is $d$-degenerate and would extend the paper's two-dimensional grid result. The conjecture remains open in general.