No pure steering bit in minimal dimension

There does not exist a family $\rho=(\rho_{a|x})_{a,x<2}$ of complex $2\times2$ matrices with all of the following properties: every block is positive semidefinite, $\sum_a\rho_{a|0}=\sum_a\rho_{a|1}$, and $\operatorname{Re}\operatorname{tr}(\tfrac12\sum_{x,a}\rho_{a|x})=1$; there exist, for each $x$, nonnegative numbers $p_{x,a}$ summing to one and vectors $v_{a,x}\in\mathbb C^2$ satisfying $\sum_i|v_{a,x,i}|^2=1$ for every $a,x$, including those with $p_{x,a}=0$, such that $\rho_{a|x}=p_{x,a}v_{a,x}v_{a,x}^\dagger$; and $\rho$ converts to every normalized family $\sigma=(\sigma_{a_f|x_f})_{a_f,x_f<2}$. Here normalized for $\sigma$ means the same positivity, equality of the two outcome sums, and unit total mass just specified for $\rho$, without any purity requirement. Conversion means that there exist a natural number $m$, matrices $K_w\in\mathbb C^{2\times2}$ with $I-\sum_{w<m}K_w^\dagger K_w$ positive semidefinite, nonnegative numbers $i_{w,x_f,x}$ with $\sum_xi_{w,x_f,x}=1$, nonnegative numbers $o_{w,a_f,a,x,x_f}$ with $\sum_{a_f}o_{w,a_f,a,x,x_f}=1$, and a real $p$ with $0<p\le1$, such that, for every $a_f,x_f$, $\sum_{w,x,a}i_{w,x_f,x}o_{w,a_f,a,x,x_f}K_w\rho_{a|x}K_w^\dagger=p\sigma_{a_f|x_f}$. These choices may depend on $\sigma$; the protocol need not be complete, and the unit vectors specifying purity need not be mutually orthogonal.