  For the last passage percolation (LPP) on $\mathbb{Z}^2$ with exponential
passage times, let $T_{n}$ denote the passage time from $(1,1)$ to $(n,n)$. We
investigate the law of iterated logarithm of the sequence $\{T_{n}\}_{n\geq
1}$; we show that $\liminf_{n\to \infty} \frac{T_{n}-4n}{n^{1/3}(\log \log
n)^{1/3}}$ almost surely converges to a deterministic negative constant and
obtain some estimates on the same. This settles a conjecture of Ledoux (J.
Theor. Probab., 2018) where a related lower bound and similar results for the
corresponding upper tail were proved. Our proof relies on a slight shift in
perspective from point-to-point passage times to considering point-to-line
passage times instead, and exploiting the correspondence of the latter to the
largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical
ingredient, which is of independent interest, is a new lower bound of lower
tail deviation probability of the largest eigenvalue of $\beta$-Laguerre
ensembles, which extends the results proved in the context of the
$\beta$-Hermite ensembles by Ledoux and Rider (Electron. J. Probab., 2010).
