The interval-maximization conjecture for translational tilings

Let $\alpha,n,d\in\mathbb{Z}^{+}$, let $C\subset\mathbb{Z}^{d}$ satisfy $|C|=n$, and let $\mathcal{T}(\alpha,C)$ denote the set of canonical translational tilings of $C$ by tiles of size $\alpha$. The interval $[n]\subset\mathbb{Z}$ is regarded as a one-dimensional contiguous set of size $n$. Interval-maximization conjecture. For every such $\alpha,n,d,C$,

$$\forall \alpha,n,d\in\mathbb{Z}^{+},\ \forall C\subset\mathbb{Z}^{d}\Big[(|C|=n)\implies|\mathcal{T}(\alpha,C)|\leq|\mathcal{T}(\alpha,[n])|\Big].$$

This is a sharper proposed form of the preceding extremal question, asserting that the interval maximizes the number of tilings among all finite sets of a given cardinality, even when the ambient dimension varies. The surrounding text presents it as a hypothesis, and the supplied material gives no resolution.