Nonnegativity conjecture for the auxiliary polynomial U_k

For each integer $k\geq1$, define

$$U_k(q)=\sum_{\ell=1}^{\lfloor k/2\rfloor}q^{2\ell(\ell+1)}\left[\genfrac{}{}{0pt}{}{k}{2\ell}\right]_q-\sum_{\ell=1}^{\lfloor k/3\rfloor}(-1)^\ell q^{\ell(9\ell+5)/2}\left[\genfrac{}{}{0pt}{}{k}{3\ell}\right]_q.$$

Here $A(q)\geq_qB(q)$ denotes coefficientwise comparison, so $U_k(q)\geq_q0$ means that all coefficients of $U_k(q)$ are nonnegative. The $U_k$ nonnegativity conjecture. For all $k\geq19$,

$$U_k(q)\geq_q0.$$

If true, this would, together with the verified range proposition, imply the $1/2$-conjecture for $r=1/3$. The claim is supported by empirical evidence and remains unproved in the source.