Continuity and strict monotonicity of the top boundary for a self-affine attractor

Let $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, and define

$$T_0(x,y)=(\lambda x,\mu y),\qquad T_1(x,y)=(\mu x+1-\mu,\lambda y+1-\lambda).$$

Let $A_{\lambda,\mu}$ be the attractor of the iterated function system $\{T_0,T_1\}$, and define its top boundary by

$$\partial_{top}(A_{\lambda,\mu})=\{(x,y)\in A_{\lambda,\mu}:\text{ for every }(x,y')\in A_{\lambda,\mu},\ y'\leq y\}.$$

Top-boundary conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $\partial_{top}(A_{\lambda,\mu})$ is the graph of a continuous, strictly increasing function.

The paper proves this for a closed subset $G$ of the parameter space occupying at least 98.3% of it, and computational evidence suggests that such sets $G$ can be made arbitrarily close to the full parameter space. The conjecture asserts the corresponding property for every admissible parameter pair.