The polynomial-factor conjecture for the number of tiles of $\mathbb{Z}^d$

For each $d$, let $t_{n,d}$ denote the number of tiles contained in $[n]^d$, and let $v_n$ denote the volume parameter used in the paper. Polynomial-factor conjecture. There is a constant $C(d)$ depending only on $d$ such that

$$t_{n,d}=O\left(n^{C(d)}\left(3^{\frac{1}{3}}\right)^{v_n}\right).$$

Moreover, one can take $C(d)=d-1$. This would sharpen the polynomial prefactor in the tile-count upper bound; the source states it as a question and provides no resolution.