= Solution
If Bob opens the trap, <Ehrenfest theorem> gives branch-dependent Newtonian accelerations
$$
a_L\simeq\frac{Gm_A}{R^2},
\qquad
a_R\simeq\frac{Gm_A}{(R-d)^2},
$$
so, for $R\gg d$,
$$
\Delta a=a_R-a_L\simeq\frac{2Gm_A d}{R^3}.
$$
After time $t$, the centers of Bob's two conditional wave packets differ by
$$
\Delta x\simeq\frac12\Delta a\,t^2
\simeq\frac{Gm_A d}{R^3}t^2.
$$
For Alice's superposition, these distinguishable Bob states become entangled with $|L\rangle$ and $|R\rangle$; for Alice's mixture there was no initial coherence to entangle. Taking Bob's minimum packet width to be the <Planck length> $\ell_P$, appreciable branch distinguishability begins when $\Delta x\sim\ell_P$, at
$$
\boxed{
t_{\rm ent}\sim
\sqrt{\frac{\ell_P R^3}{Gm_A d}}}.
$$
Keeping the trap closed suppresses this branch separation.
If $t_{\rm ent}<R/c$, Bob could choose whether to destroy Alice's local coherence before a light signal from his laboratory arrived. Any procedure by which Alice distinguished the coherent superposition from the mixture in less than $R/c$ would then enable superluminal signalling. The largest dangerous separation is determined parametrically by $t_{\rm ent}\sim R/c$, which gives
$$
R\sim\frac{Gm_A d}{c^2\ell_P}.
$$
Causality therefore requires
$$
T_A\gtrsim\frac Rc
\sim\frac{Gm_A d}{c^3\ell_P}.
$$
Using $Gm_P/c^2=\ell_P$ for the <Planck mass> $m_P$,
$$
\boxed{
T_A\gtrsim
\frac{m_A}{m_P}\frac dc}
$$
up to numerical factors.
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