TITLE: Strong chains in $[\omega_2]^{\omega_2}$ mod finite of length $\omega_3$
QUESTION [5 upvotes]: Probing a bit the difference between $[\omega_1]^{\omega_1}$ and $[\omega_2]^{\omega_2}$ modulo the finite sets:

Question
Can there exist a family $\langle X_\alpha:\alpha<\omega_3\rangle$ of
sets in $[\omega_2]^{\omega_2}$ such that for $\alpha<\beta<\omega_3$,

$|X_\beta\setminus X_\alpha|=\omega_2$, and
$ X_\alpha\setminus X_\beta$ is finite?


The existence of such a family (a strong chain of length $\omega_3$ in $[\omega_2]^{\omega_2}$ mod finite) implies the existence of a strongly almost disjoint family in $[\omega_2]^{\omega_2}$ of size $\omega_3$:  let $A_\alpha:=X_{\alpha+1}\setminus X_\alpha$, and we get a collection of size $\omega_3$ in $[\omega_2]^{\omega_2}$ with pairwise finite intersection.  The existence of such a family is consistent, as shown by Baumgartner [1].
Koszmider [2] showed that you can have long chains (of length $\omega_2$) in $[\omega_1]^{\omega_1}$ modulo the finite ideal, so this question is an attempt to lift his result to $[\omega_2]^{\omega_2}$.  (I do not even know the answer for $[\omega_2]^{\omega_2}$ if ``finite'' is replaced by countable, but obviously a consistency result there should be easier to obtain, I just don't know if it's been done.)
There are differences between $\omega_1$ and $\omega_2$ that indicate this question may be interesting.  Koszmider [3] showed that one can have a sequence $\langle f_\alpha:\alpha<\omega_2\rangle$ that is strictly increasing mod finite, while Shelah [4] has shown in ZFC that this phenomenon CANNOT happen at $\omega_2$: there is no sequence $\langle f_\alpha:\alpha<\omega_3\rangle$ of functions in $^{\omega_2}\omega_2$ that is strictly increasing mod finite.
[1] Baumgartner, James E., Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9, 401-439 (1976). ZBL0339.04003.
[2] Koszmider, Piotr, On the existence of strong chains in $(\wp(\omega_1)/ \text{Fin})$, J. Symb. Log. 63, No. 3, 1055-1060 (1998). ZBL0936.03043.
[3] Koszmider, Piotr, On strong chains of uncountable functions, Isr. J. Math. 118, 289-315 (2000). ZBL0961.03039.
[4] Shelah, Saharon, On long increasing chains modulo flat ideals, Math. Log. Q. 56, No. 4, 397-399 (2010). ZBL1200.03031.

REPLY [3 votes]: No, there can't. The argument is not too different from Shelah's. I emailed you a note.