Homological stability for stabilized topological chiral homology

Let $M$ be an open connected parallelizable $n$-manifold, and let $A$ be an $E_n$-algebra with $\pi_0(A)=\mathbb{N}_0$. Fix $b\in A_1$, and let

$$t_b:(\int_M A)_k\longrightarrow(\int_M A)_{k+1}$$

be a stabilization map. Say that $A$ has homological stability if there is a function $r:\mathbb{N}_0\to\mathbb{N}_0$ tending to infinity such that multiplication by $b$ induces an isomorphism

$$m_{b*}:H_i(A_k)\longrightarrow H_i(A_{k+1})$$

for $i\leq r(k)$.

Homological stability conjecture. If $A$ has homological stability, then there is a range of dimensions tending to infinity below which $t_b$ induces an isomorphism on homology. Moreover, this range should depend only on the homological stability range of $A$.

This conjecture asks whether homological stability for the labels implies homological stability for the stabilized topological chiral homology of an open parallelizable manifold. The source does not specify the function describing the resulting range or establish the assertion.