Almost everywhere convergence of orthogonal series

Question

If $h_0, h_1 h_2, \ldots$ is an orthogonal sequence in the Hilbert space $L^2 (X, \Omega, \mu)$ such that $\sum\limits_{n = 0}^{\infty} \lVert h_n\rVert_2^2 < + \infty$, then does $\sum\limits_{n = 0}^\infty h_n$ converge almost everywhere?

It is easy to see that $\sum\limits_{n = 0}^\infty h_n$ converges in $L^2 (\mu)$ by the Riesz-Fischer theorem, and also that there is a 'subseries' that converges almost everywhere, i.e. for some increasing sequence $N_0 < N_1 < N_2 < \ldots$, the sequence $S_n = \sum\limits_{k = 0}^{N_n} h_{k}$ converges almost everywhere. But can we assert anything about the original series?

Answer 1

Observe that there exists a subsequence $$ \sigma_{n_k} := \sum_{j=0}^{n_k} |h_j| $$ that converges almost everywhere in $X$. But this implies that the series with non-negative terms $\sum_j |h_j|$ converges almost everywhere.

EDIT: under the current assumptions ($\sum_j \|h_j\|_2^2 < \infty$) this answer is no longer correct.

Answer 2

I doubt that it's true, not that I have a counterexample handy. For many years people thought this was false for trigonometric series; Carleson astounded everyone by showing that the answer is yes for trigonometric series. but this is like the hardest theorem in harmonic analysis.