  We provide a near-complete classification of the Lorentz spaces
$\Lambda_{\varphi}$ for which the sequence $\{S_{n}\}_{n\in \mathbb{N}}$ of
partial Fourier sums is almost everywhere convergent along lacunary
subsequences. Moreover, under mild assumptions on the fundamental function
$\varphi$, we identify $\Lambda_{\varphi}:= L\log\log L\log\log\log\log L$ as
the \emph{largest} Lorentz space on which the lacunary Carleson operator is
bounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecture
stated by Konyagin in his 2006 ICM address. Our proof relies on a newly
introduced concept of a "Cantor Multi-tower Embedding," a special geometric
configuration of tiles that can arise within the time-frequency tile
decomposition of the Carleson operator. This geometric structure plays an
important role in the behavior of Fourier series near $L^1$, being responsible
for the unboundedness of the weak-$L^1$ norm of a "grand maximal counting
function" associated with the mass levels.
