Bob-to-Alice 1W-LOCC preserves unsteerability

Let $d,r,s,d_f,r_f,s_f,m$ be natural numbers and let $P$ be a protocol whose data require $d,r,s,d_f,r_f,s_f>0$, matrices $K_w\in\mathbb C^{d_f\times d}$ for $w<m$ satisfying $I_d-\sum_wK_w^\dagger K_w\succeq0$, input probabilities $\alpha_w(x\mid x_f)$ summing to $1$ over $x<s$ for every $w,x_f$, and output probabilities $\beta_w(a_f\mid a,x,x_f)$ summing to $1$ over $a_f<r_f$ for every $w,a,x,x_f$. For a matrix family $F$ with $r$ outcomes and $s$ settings, put $B(F)=s^{-1}\sum_x\sum_aF_{a,x}$ and $\mu(F)=\operatorname{Re}\operatorname{tr}B(F)$. Suppose $\rho$ has positive semidefinite blocks, setting-independent outcome sums, mass $\mu(\rho)\in[0,1]$, and a finite hidden-state representation

where $h\in\mathbb N$, $p_\ell$ and each $q_{\ell,x}$ are nonnegative probability distributions, and every $\xi_\ell$ is positive semidefinite with trace $1$. Then the output family

also has positive semidefinite blocks, setting-independent outcome sums, mass in $[0,1]$, and a representation of the same finite hidden-state form using its own mass. Moreover, if its mass $\mu(R)=\operatorname{Re}\operatorname{tr}(s_f^{-1}\sum_{x_f,a_f}R_{a_f,x_f})$ is positive, then the family $R_{a_f,x_f}/\mu(R)$ has mass $1$, satisfies those assemblage conditions, and has a finite hidden-state representation. Completeness $\sum_wK_w^\dagger K_w=I_d$ is not assumed; $m=0$ is allowed and gives zero output, for which the positive-mass implication is vacuous.