The congruence-kernel conjecture for super-modular categories

Let $\mathcal{B}$ be a super-modular category of rank $2k$, and let $\hat{S}$ and $\hat{T}^2$ be the corresponding matrices. Let $\Gamma_\theta=\langle\mathfrak{s},\mathfrak{t}^2\rangle<\operatorname{SL}(2,\mathbb{Z})$ be the spin mapping class group of the torus with spin structure $(A,A)$, and let $\pi_k$ denote passage to $\operatorname{PSU}(k)$. The congruence-kernel conjecture. The projective representation

$$\hat{\rho}:\Gamma_\theta\longrightarrow \operatorname{PSU}(k),\qquad \hat{\rho}(\mathfrak{s})=\pi_k(\hat{S}),\quad \hat{\rho}(\mathfrak{t}^2)=\pi_k(\hat{T}^2)$$

has kernel a congruence subgroup. If true, the image $\hat{\rho}(\Gamma_\theta)$ is finite; the level of the kernel is not known in general.