= Solution
Let the bunch tail have the higher <energy> so that it catches the head. For an ultrarelativistic <proton>, $\delta v/v\simeq(\delta E/E)/(\beta^2\gamma^2)$. A drift of <length> $S$ then changes the bunch <length> by approximately $S\delta v/v$. Requiring that to match the initial <length> gives
$$
\boxed{\frac{\Delta E}{E}\simeq\beta^2\gamma^2\frac{\ell_p}{S_{\rm prop}}.}
$$
Taking $\gamma\simeq62.5/0.938\simeq66.6$, $\beta\simeq1$, and the given <lengths> yields $\Delta E/E\simeq0.023$, about $2.3\%$. Treating $62.5$ GeV as kinetic rather than total <energy> changes that by only a few percent.
This is the linear ballistic compression requirement; a real finite-emittance bunch, nonlinear velocity-energy relation, uncorrelated <energy> spread and collective fields prevent exact zero <length>. The chirp is a correlated full head-to-tail spread, not automatically an rms <energy> spread.
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