The Mersenne-type binary polynomial odd-sequence length conjecture

Let $M_1=x^2+x+1$, and let the odd sequence of a polynomial be the sequence of odd polynomials obtained by the binary polynomial Collatz transformations. Let $r\geq 1$ and $0\leq j\leq 2^{r-1}-1$, and set

$$A=M_1^{2^r-j}+1.$$

Odd-sequence length conjecture. The length of the odd sequence of $A$ is $j+1$. The claim concerns the special family $M_1^n+1$ and is supported by the examples and computations displayed in the source, but remains unproved there.