Crossing-number conjecture for consecutive Chebyshev degrees

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. For an odd integer $i\ge 3$, the consecutive-degree crossing-number conjecture. The parametrization

$$(T_i,T_{i+1},T_{i+2})$$

parametrizes a knot with crossing number $i$. The source says this pattern has been verified through a finite range in a related family but reports no proof for all values, and presents the assertion as an open problem.