= Solution
At conjunction, the planetesimal's <mean anomaly> is
$$
M_c=\lambda_c-\varpi
=\frac{\phi_1+2\pi k}{3},
\qquad k=0,1,2.
$$
If $\phi_1=\pi+\delta$ with $|\delta|\leq\Delta\phi_1\leq\pi$, the conjunction branch that can come closest to periapsis has
$$
\boxed{|M_c|_{\min}=\frac{\pi-\Delta\phi_1}{3}}.
$$
Let $E_{min}$ solve <Kepler's equation>
$$
E_{min}-e\sin E_{min}=\frac{\pi-\Delta\phi_1}{3}.
$$
At that phase the orbital radius is $r_{min,c}=a(1-e\cos E_{min})$. A geometrical close encounter is possible only if
$$
\boxed{a(1-e\cos E_{min})\lesssim a_1+R_{\rm enc}},
$$
where $R_{\rm enc}$ may be chosen as the planet's <Hill radius> or another encounter distance. With a point planet, set $R_{\rm enc}=0$. This implicit inequality is the requested eccentricity constraint as a function of $\Delta\phi_1$.
The weaker necessary condition that the orbits cross is
$$
\boxed{e\geq1-\frac{a_1}{a}
=1-\left(\frac58\right)^{2/3}}.
$$
It becomes sufficient for phase access only as $\Delta\phi_1\to\pi$, when a conjunction can approach periapsis. Smaller libration amplitude keeps conjunctions farther from periapsis and requires a larger eccentricity than this orbit-crossing bound.
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