$H\cdot(109,-3)$ has no even-only SAFE extension containing a new entry

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 $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$. 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. $Q$ is SAFE if it is $\theta$-safe for some tree witness $\theta$. Let $H_1=(0,1,-1,5,3,2,-15,-11,-5,-99,-115,-55,109,-3)$, which is SAFE. If $D$ is a finite word of even integers, at least one of which does not occur in $H_1$, then $H_1\cdot D$ is not SAFE. (The new-entry condition is necessary: appending already present letters does not change SAFE-ness.) This is GPT Astra's preparation obstruction: any SAFE continuation of $H_1$ with a new even entry must also add new odd entries.