Monotonicity conjecture for restricted graphical Stirling numbers

Let $G$ be a graph, let $k$ be fixed, and let $\st{G}{k}_r$ denote the $r$-restricted graphical Stirling number, where the restricted-set size is indexed by $r$. Restricted-number monotonicity conjecture. For any graph $G$ and fixed $k$, the restricted graphical Stirling numbers satisfy

$$\st{G}{k}_1\geq\st{G}{k}_2\geq\dots\geq\st{G}{k}_k.$$

Equivalently, the sequence decreases strictly as the size of the restricted set increases.

The conjecture proposes monotonicity for every graph and fixed $k$; no supporting cases or resolution are supplied in the source.