Filtration monotonicity under refinement for canonical relation lattices

Let $N$ be the common lattice of a fan $\Sigma$ and its subdivision $\Sigma'$, with $\Sigma(1)\subseteq\Sigma'(1)$. Write $L_{\mathrm{rel}}(\Sigma)$ and $L_{\mathrm{rel}}(\Sigma')$ for their relation lattices, and let

$$j:L_{\mathrm{rel}}(\Sigma)\hookrightarrow L_{\mathrm{rel}}(\Sigma')$$

be the natural injection induced by $\mathbb{Z}^{\Sigma(1)}\hookrightarrow\mathbb{Z}^{\Sigma'(1)}$. For $0\neq r\in L_{\mathrm{rel}}(\Sigma)$, define its filtration depth by

$$\operatorname{depth}_\Sigma(r):=\min\{k\mid r\in F_kL_{\mathrm{rel}}(\Sigma)\}.$$

Filtration monotonicity under refinement. One has

$$\operatorname{depth}_{\Sigma'}(j(r))\le \operatorname{depth}_{\Sigma}(r).$$

Equivalently, subdivision can only preserve or lower the minimal filtration level needed to express a relation using star-supported generators. This predicts that refinement does not make relations more complicated with respect to the star-support filtration; whether the assertion holds in full generality remains open in the supplied source.