Let $k$ be a field of characteristic zero. It is well-known that if $(x,y) \mapsto (p,q) \in k[x,y]^2$ is a counterexample to the two-dimensional Jacobian Conjecture, then there exists an automorphism $g$ of $k[x,y]$ such that the support of $g(p)$ is contained in some rectangular $\{ (i,j) | 0 \leq i \leq a, 0 \leq j \leq b \}$ with $(a,b)$ in the support of $g(p)$; see Corollary 10.2.21 of van den Esses's book.
Question 1 (easy?): Is it true that in that case we also have: $g(q)$ is contained in some rectangular $\{(i,j) | 0 \leq i \leq c, 0 \leq j \leq d\}$ with $(c,d)$ in the support of $g(q)$?
By Proposition 2.1 (with Proposition 1.13): $\mu x^cy^d$ is the $(1,1)$-leading term of $g(q)$, for some $\mu\ \in k-\{0\}, c,d \in \mathbb{N}$, since the $(1,1)$-leading term of $g(p)$ is $\lambda x^ay^b$, for some $\lambda \in k-\{0\}, a,b \in \mathbb{N}$. Hence, $(c,d)$ is in the support of $g(q)$.
But should $g(q)$ be contained in some rectangular?
Remark: Perhaps the arguments in the proof of Corllary 10.2.21 of Essen's book actually prove that $g(q)$ must also be subrectangular? (the argument about no edge with a negative slope).
Interestingly, in the non-commutative case, namely, in the first Weyl algebra, the answer is yes, see Theorem 5.12.
Question 2 (more difficult?): Is it true that we can further obtain the a possible counterexample has the form: $g(p)$ is as above and, in addition, $\{(a,v)_{0 \leq v \leq b-1}\}$ are not in the support of $g(p)$ and $\{(u,b)_{0 \leq u \leq a-1}\}$ are not in the support of $g(p)$? Something like the picture of $(p_0,q_0)$ on page 50.
I think that I have once seen a half positive answer to my Question 2, see the picture on page 21 (there: both $P$ and $\phi(P)$ are half good for me).
Edit: My question 1 has a positive answer thanks to Lemma B or Theorem 3.4.
Now I also asked Question 2 in MO.
Question 1: yes
Question 2: no
All you can say is that if $a<b$, then you can assume that (u,b) is not in the support of g(p) for $u<a$. See for example Makar-Limanov's "On the Newton polygon of a Jacobian mate" at
https://www.mpim-bonn.mpg.de/preprints
the preprint 2013-53.
I think that the picture you want to use in order to prove the positive answer to question 2, is only a particular case with (a,b)=m(4,12).