Infinitely many inequivalent minimal-dimensional steering classes

For each real $\theta$, define a family of $2\times2$ matrices by $A^\theta_{a|x}=\tfrac12v^\theta_{a,x}(v^\theta_{a,x})^\dagger$, for $a,x\in\{0,1\}$, where $v^\theta_{0,0}=(1,0)$, $v^\theta_{1,0}=(0,1)$, $v^\theta_{0,1}=(\cos\theta,\sin\theta)$, and $v^\theta_{1,1}=(-\sin\theta,\cos\theta)$. A family $\rho_{a|x}$ is normalized here when all its blocks are positive semidefinite, $\sum_a\rho_{a|0}=\sum_a\rho_{a|1}$, and $\operatorname{Re}\operatorname{tr}(\tfrac12\sum_{x,a}\rho_{a|x})=1$. It is pure orthogonal when there exist probability vectors $(p_{x,a})_{a<2}$ and vectors $u_{a,x}\in\mathbb C^2$ with $\sum_i\overline{u_{a,x,i}}u_{b,x,i}=1$ if $a=b$ and zero otherwise, such that $\rho_{a|x}=p_{x,a}u_{a,x}u_{a,x}^\dagger$. A normalized family has a finite local hidden-state model when, for some natural number $h$, it can be written $\rho_{a|x}=\sum_{\ell<h}p_\ell q_{\ell,x,a}\xi_\ell$, where $p_\ell\ge0$, $\sum_\ell p_\ell=1$, each $\xi_\ell$ is positive semidefinite with trace one, and each $q_{\ell,x,a}$ is nonnegative with $\sum_aq_{\ell,x,a}=1$. Conversion from a normalized family $\rho$ to a normalized family $\sigma$ means that there exist a natural number $m$, matrices $K_w\in\mathbb C^{2\times2}$ with $I-\sum_wK_w^\dagger K_w$ positive semidefinite, probability vectors $i_{w,x_f,x}$ over $x$, probability vectors $o_{w,a_f,a,x,x_f}$ over $a_f$, and a real $p$ with $0<p\le1$, such that $\sum_{w,x,a}i_{w,x_f,x}o_{w,a_f,a,x,x_f}K_w\rho_{a|x}K_w^\dagger=p\sigma_{a_f|x_f}$ for every $a_f,x_f$; completeness of the matrices $K_w$ is not required. The declaration asserts that every $A^\theta$ with $0<\theta<\pi/4$ is normalized and pure orthogonal and has no finite local hidden-state model; for any distinct $\theta,\eta$ both in that open interval, neither conversion $A^\theta\to A^\eta$ nor conversion $A^\eta\to A^\theta$ exists; and the set $\{A^\theta:0<\theta<\pi/4\}$ of actual matrix families is infinite.