Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two polynomials $f(q)$ and $g(q)$, we write $f(q)\geq g(q)$ provided that $f(q)-g(q)$ is a polynomial having non-negative coefficients.

I would like to ask:

QUESTION.Suppose $0\leq k\leq a<b$ are integers. Is it true that $\binom{b+a}{b-k}_q\geq\binom{a+b}{a-k}_q$?