Root conflict for $P_n\cdot T$: tree-safe words meet $\{2N+2,3N+3,2T-1,3T-2\}$

A tree witness is a function $\theta(k,r)\in\{0,1\}$ on the nodes $(k,r)$, $0\le r<2^k$, of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ (two's-complement digits). A finite word $Q$ of integers is $\theta$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression. Let $n\ge1$, $N=2^n$ and $T=4N$. In every $\theta$-safe word $Q$ (any tree witness $\theta$) containing $0,1,N+1,T$ in which $0$ occurs before $N+1$ and $1$ occurs before $T$,

because the open pairs $(0,N+1)$ and $(1,T)$ would force opposite values at the root node. In particular every tree-safe extension of $P_n=\{0,\dots,N-1\}$ (in any order) that contains $N+1$ after its prefix and $T$ contains one of these four numbers: the fourth-term repair of the family $P_n\cdot T$ fails (Erdős Problem 195).

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. "Begins with" is the list prefix relation <+:; positions in a word are List.idxOf. The statement quantifies over every witness θ, so it holds in particular for all ε-orders.