The exact factor-complexity formula for the $(3/2)^n$ steering word

Let $T=(t_n)_{n\ge0}$ be the steering word considered in the paper, and let $p_T(k)$ denote the number of distinct factors of length $k$ occurring in $T$. Exact complexity conjecture.

$$p_T(k)=3^{\,k+1}-2^{\,k+1}\quad\text{for every }k\ge1.$$

The formula agrees with the measured complexity profile for small $k$ and predicts the exact number of factors at every length. The paper presents it as conjectural; in particular, the available computation is finite and does not establish the assertion for all $k$.