In a previous question, I asked about the impact of strong chains in $^{\omega_1}\omega_1$ (e.g., sequences of functions $\langle f_\alpha:\alpha<\kappa\rangle$ in $^{\omega_1}\omega_1$ that are strictly increasing modulo the ideal of finite sets) on cardinal characteristics of the continuum.
Koszmider [1] proved it is consistent that there is a strong chain in $^{\omega_1}{\omega_1}$ of length $\omega_2$, and Veličković and Venturi showed that Neeman's techniques (finite approximations with side conditions consisting of elementary submodels of two types) can be adapted to obtain the same result. Certainly this latter proof does not adapt in a straightforward manner to yield chains of even longer length. I am less familiar with Koszmider's original proof, but that seems to work only for $\omega_2$ as well.
Question: Is it consistent that there is a sequence $\langle f_\alpha:\alpha<\omega_3\rangle$ in $^{\omega_1}\omega_1$ that is increasing modulo the ideal of finite sets? What is known about this?
[1] Koszmider, Piotr, On strong chains of uncountable functions, Isr. J. Math. 118, 289-315 (2000). ZBL0961.03039.
[2] Veličković, Boban; Venturi, Giorgio, Proper forcing remastered, Cummings, James (ed.) et al., Appalachian set theory 2006–2012. Based on the Appalachian set theory workshop series during the period 2006–2012. Cambridge: Cambridge University Press (ISBN 978-1-107-60850-4/pbk). London Mathematical Society Lecture Note Series 406, 331-362 (2013). ZBL1367.03094.
