Jungers–Protasov–Blondel conjecture on the overlap-free-word growth exponents

Let $A_1,A_2$ be the special $20\times20$ matrices with nonnegative integer entries associated with overlap-free words, and let $\mathcal{M}=\{A_1,A_2\}$. Let $\beta$ be the upper exponent of growth of the number of overlap-free binary words, and let $\rho$ denote the spectral radius. Jungers–Protasov–Blondel conjecture. The s.m.p. for the family $\mathcal{M}$ is $A_1A_2$, and

$$\beta=\frac{1}{2}\log_2\rho(A_1A_2).$$

The conjecture gives an exact value for the upper growth exponent through a spectrum-maximizing product; the paper reports only numerical bounds for $\beta$, so the assertion remains unresolved in the supplied source.