  Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the
family of random distance functions $\{D_h^\epsilon\}_{\epsilon >0}$ on the
plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where
$\{h_\epsilon\}_{\epsilon >0}$ is a smooth mollification of the planar Gaussian
free field. Recent works have shown that for all $\xi > 0$ the LFPP metrics,
appropriately re-scaled, admit non-trivial subsequential limiting metrics. In
the case when $\xi < \xi_{\mathrm{crit}} \approx 0.41$, it has been shown that
the subsequential limit is unique and defines a metric on $\gamma$-Liouville
quantum gravity $\gamma = \gamma(\xi) \in (0,2)$.
  We prove that for all $\xi > 0$, each possible subsequential limiting metric
is nearly bi-Lipschitz equivalent to the LFPP metric $D_h^\epsilon$ when
$\epsilon$ is small, even if $\epsilon$ does not belong to the appropriate
subsequence. Using this result, we obtain bounds for the scaling constants for
LFPP which are sharp up to polylogarithmic factors. We also prove that any two
subsequential limiting metrics are bi-Lipschitz equivalent.
  Our results are an input in subsequent work which shows that the
subsequential limits of LFPP induce the same topology as the Euclidean metric
when $\xi = \xi_{\mathrm{crit}}$. We expect that our results will also have
several other uses, e.g., in the proof that the subsequential limit is unique
when $\xi \geq \xi_{\mathrm{crit}}$.
