This is a follow up from my earlier MO question.

Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its conjugate partition $\lambda'$. Denote by $\lambda''=\lambda',0$ found by appending one extra zero at the right end of $\lambda'$. Further, define the following numeric $b(\lambda'')=\#\{j: \lambda_j''-\lambda_{j+1}''>0\}$.

For example, if $\lambda=(4,2,1)$ then $\lambda'=(3,2,1,1)$ and $\lambda''=(3,2,1,1,0)$ and $b(\lambda'')=3$.

Consider the polynomials $$f_n(q):=\sum_{\lambda\vdash n}(q-1)^{b(\lambda'')-1}\,q^{\ell(\lambda)-b(\lambda'')}.\tag1$$

Denote by $t_n$ the largest $t$ such that $q^t$ divides $f_n(q)$.

QUESTION 1.Is it true that $t_n\in\{0,1,2\}$?

QUESTION 2.(stronger) Is it true that the "infinite word" $\,t_1t_2t_3\cdots=0\prod_{k=1}^{\infty}01^{2k}02^k$?