${\mathrm{S}\mathrm{L}}_2$-character varieties of $2$-generated groups and failure of weak integrality

Let $\mathrm{\ell }$ be a prime number, $k$ a positive integer and consider the group ${\mathrm{\Gamma }}_{{\mathrm{\ell }}^k}:=⟨a,b\mathrm{\mid }{a}^{{\mathrm{\ell }}^k({\mathrm{\ell }}^k-1)}b{a}^{-{\mathrm{\ell }}^k}{b}^{-2}⟩$. We prove that ${\mathrm{\Gamma }}_{{\mathrm{\ell }}^k}$ is not ${\mathrm{S}\mathrm{L}}_2$-weakly integral with obstruction at exactly the prime $\mathrm{\ell }$. We also give a general description of the character varieties of $2$-generated groups with a relation of the form $a^{n_1}{b}^{m_1}{a}^{n_2}{b}^{m_2}=1$.