Protasov's conjecture on spectrum-maximizing and spectrum-minimizing products

Let $r$ be an odd integer, and let $A_1,A_2$ be the $(r-1)\times(r-1)$ matrices defined by

$$(A_s)_{ij}=1\quad\text{if }2-s\leq 2j-i\leq r-s+1,$$

and $(A_s)_{ij}=0$ otherwise, for $s=1,2$. A product is an s.m.p. if it is a spectrum-maximizing product, and an s.l.p. if it is a spectrum-minimizing product. Protasov's conjecture. For every odd $r$, one of the two products $A_1$ and $A_1A_2$ is an s.m.p. and the other is an s.l.p. The conjecture concerns exact extremal products for the joint and lower spectral radii of this matrix family; the supplied source states it as unresolved.