The $q$-Pochhammer symbol
$(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$.
The $q$-binomial coefficients (also known as the Gaussian polynomials)
$$
\genfrac{[}{]}{0pt}{}{m}{n}
={\genfrac{[}{]}{0pt}{}{m}{n}} _q
:=\frac{(q) _ m}{(q) _ n(q) _ {m-n}}
$$
(zero when $n<0$ or $n>m$) are polynomials as well, of degree $n(m-n)$, in fact with
all coefficients positive. The latter circumstance is a source of many other positivity
(at least nonnegativity) claims like for the 2-parameter polynomial family
$$
V_{n,k}(q)
:=\sum_{j=0}^nq^{j^2+kj}\genfrac{[}{]}{0pt}{}{n+k-j}{k}
\sum_{m=0}^{\min\lbrace j,n+k-j\rbrace}q^{m^2+km}
\genfrac{[}{]}{0pt}{}{n+k-j}{m}\genfrac{[}{]}{0pt}{}{n+k}{j-m}.
$$
My problem is the *expected nonnegativity of another family*
$$
U_{n,k,t}(q)
:=\sum_{j=0}^nq^{j^2+(k+t)j}\genfrac{[}{]}{0pt}{}{n+k-j}{k}
\sum_mq^{m^2+(k+t)m}\genfrac{[}{]}{0pt}{}{2n+2k-j-m}{j-m}\genfrac{[}{]}{0pt}{}{n+k-j}{m}\genfrac{[}{]}{0pt}{}{n-j}{m}(q)_m
$$
(note the unpleasant appearance of the $q$-Pochhammer symbol at the end) *when* $t=k$.
The parameter $t$ is introduced by purpose, as it gives more flexibility to the
polynomial family, namely, $U_{n,k,0}(q)=V_{m,k}(q)$ (a corollary of a known
hypergeometric identity) is positive and
$$
U_{n,k,t}(q)
=q^nU_{n,k,t-1}(q)+(1-q^{k+1})U_{n-1,k+1,t-1}(q),
$$
which clearly lacks of positivity because of the factor $1-q^{k+1}$.
Experimental verification shows that the polynomials $U_{n,k,t}(q)$ have only nonnegative
coefficients for the range $t=0,1,\dots,t_0$ where $t_0\ge2k+1$.
Again, even the particular case $t=k>0$ is out of my reach.

The problem is related to the positivity issues from my earlier problem, although it will require a definite text to make the relation clear.