Diagonal monotonicity conjecture for kicked Harper rotation sets

Let $F_{\alpha,\beta}$ denote the kicked Harper map with parameters $(\alpha,\beta)$, and let $\rho(F_{\alpha,\beta})$ denote its rotation set. For parameters on the diagonal, consider $F_{\alpha,\alpha}$.

Diagonal monotonicity conjecture. If $0\leq\alpha\leq\tilde\alpha$, then

$$\rho(F_{\alpha,\alpha})\subseteq\rho(F_{\tilde\alpha,\tilde\alpha}).$$

The conjecture is motivated by numerical simulations suggesting monotonic behavior along the diagonal, despite the failure of monotonicity for general parameter changes. The source gives no resolution, so the conjecture remains open.