This is a follow up on 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 two numerics $a(\lambda'')_j=\lambda_j''-\lambda_{j+1}''$ for $j=1,2,\dots,\ell(\lambda')$.

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

QUESTION. Is it true that the coefficients of the polynomial $A_n(q)$ are all in $\{-1,0,1,2\}$? $$A_n(q):=\sum_{\lambda\vdash n}q^{n-\ell(\lambda)} \prod_{a(\lambda'')_j\geq1}\frac{(q^{2a(\lambda'')_j}-1)(q-1)}{q+1}.$$

REMARK. In fact, it appears that only coefficient of the middle-term can possibly be equal to $2$.

This is a follow up on 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 $\tilde\lambda'=\lambda,0$ found by appending one extra zero at the right end of $\lambda$. Further, define the following numerics $a(\lambda)_j=\tilde\lambda_j-\tilde\lambda_{j+1}$ for $j=1,2,\dots,\ell(\lambda)$.

For example, if $\lambda=(3,2,1,1)$ and $\tilde\lambda=(3,2,1,1,0)$ and $a(\lambda)=(1,1,0,1)$.

QUESTION. Is it true that the coefficients of the polynomial $A_n(q)$ are all in $\{-1,0,1,2\}$? $$A_n(q):=\sum_{\lambda\vdash n}q^{n-\lambda_1} \prod_{a(\lambda)_j\geq1}\frac{(q^{2a(\lambda)_j}-1)(q-1)}{q+1}.$$

REMARK. In fact, it appears that only coefficient of the middle-term can possibly be equal to $2$.