Gukov-Pei-Putrov-Vafa conjecture for $SU(N)\mathrm{/}{\mathbb{Z}}_m$

In our earlier work, we studied the $\widehat{Z}$-invariant(or homological blocks) for $SO(3)$ gauge group and we found it to be same as ${\widehat{Z}}^{SU(2)}$. This motivated us to study the $\widehat{Z}$-invariant for quotient groups $SU(N)\mathrm{/}{\mathbb{Z}}_m$, where $m$ is some divisor of $N$. Interestingly, we find that $\widehat{Z}$-invariant is independent of $m$.