Longest arithmetic progressions in generalised Thue–Morse sequences

Let $p,q,n\in\mathbb{N}^{+}$, set $Q=p+q$, and let $A_{p,q}(d)$ denote the maximum length of an arithmetic progression of difference $d$ occurring in the generalised Thue–Morse sequence associated with parameters $p,q$. Assume $n>2$. Longest-progression conjecture.

$$\max_{d\leqslant Q^n+1}A_{p,q}(d)=\begin{cases} A_{p,q}(Q^n-1)=Q^n+Q+2, & \text{if }p=q=1\text{ and }n\text{ is even},\\ A_{p,q}(Q^n-1)=Q^n+Q, & \text{if }p=q>1\text{ and }n\text{ is even},\\ A_{p,q}(Q^n+1)=Q^n+Q, & \text{if }p=q=1\text{ and }n\text{ is odd},\\ A_{p,q}(Q^n+1)=Q^n+Q-1, & \text{if }q>p=1\text{ or }p>q=1,\\ A_{p,q}(Q^n+1)=Q^n+Q-2, & \text{if }p,q>1\text{ and }p\ne q\text{ or }n\text{ is odd}.\end{cases}$$

The preceding results establish the relevant long progressions for differences $Q^n+1$, and for differences $Q^n-1$ when $p=q$; the conjecture asserts that no longer progression occurs for any difference up to $Q^n+1$.