Equality of spectral-radius limits for generalized affine fractal interpolation functions

Let $I=[0,1]$ be the domain, let $S$ be the vertical scaling function, and let $M_k$ and $M'_k$ be the matrices associated with the $k$-th subdivision, with spectral radii denoted by $\rho(M_k)$ and $\rho(M'_k)$, respectively. Assume that the vertical scaling function $S$ is not identically zero on every subinterval of $I$. Spectral-radius limit conjecture. Then

$$\lim_{k\to \infty}\rho(M_k)=\lim_{k\to \infty}\rho(M'_k).$$

The equality would identify the limiting spectral radius obtained from the upper and lower matrix approximations, supporting the spectral-radius formula used to determine the box dimension of the generalized affine fractal interpolation function. The supplied text does not indicate whether this assertion has been proved or disproved.