Monotonicity conjecture for the optimal regulation measure

Let $\bar{\eta}_1$ and $\bar{\eta}_2$ be cumulative-effort constraints satisfying $0<\bar{\eta}_1<\bar{\eta}_2$, and let $\mu_{\alpha_{opt},\bar{\eta}_i}$ be the measures associated with the corresponding optimizing solutions $\alpha_{opt,\bar{\eta}_i}$. For $t<s$, write $\mu_{\alpha_{opt},\bar{\eta}_i}([t,s))=\alpha_{opt,\bar{\eta}_i}(s)-\alpha_{opt,\bar{\eta}_i}(t)$. Monotonicity conjecture. The measure $\mu_{\alpha_{opt},\bar{\eta}_2}$ dominates $\mu_{\alpha_{opt},\bar{\eta}_1}$ in the sense that

$$\mu_{\alpha_{opt},\bar{\eta}_1}([t,s))\leq\mu_{\alpha_{opt},\bar{\eta}_2}([t,s)).$$

Equivalently,

$$\alpha_{opt,\bar{\eta}_1}(s)-\alpha_{opt,\bar{\eta}_1}(t)<\alpha_{opt,\bar{\eta}_2}(s)-\alpha_{opt,\bar{\eta}_2}(t).$$

The conjecture would provide a unified explanation for the numerically observed monotonicity of the support, the absolutely continuous density, and the atomic masses of the optimizing measure as the cumulative-effort constraint increases; these observations are not yet proved by the analytical results.