Kolmogorov rearrangement problem and Garsia's conjecture

Let $(X,\mu)$ be a probability space and let $(\varphi_n)_{n\ge 1}$ be a complete uniformly bounded orthonormal system in $L^2(X,\mu)$, meaning that $\sup_n\lVert\varphi_n\rVert_{\infty}<\infty$. The rearrangement assertion asks whether there always exists a permutation $\pi$ of $\mathbb{N}$ such that, for every coefficient sequence $(a_n)\in\ell^2$, the series $\sum_{n=1}^{\infty}a_n\varphi_{\pi(n)}(x)$ converges for $\mu$-almost every $x\in X$. Equivalently, the claimed counterexample is a complete uniformly bounded orthonormal system for which, for every permutation $\pi$, there exists $(a_n)\in\ell^2$ such that $\sum_{n=1}^{\infty}a_n\varphi_{\pi(n)}(x)$ diverges for almost every $x$.