Random-phase triangular factorization conjecture for loops in $SU(2)$

Let $k_2$ be the loop associated with a sequence $\zeta=(\zeta_n)$ satisfying $\zeta\in l^2$, and suppose the phases of the $\zeta_k$ are uniform and independent random variables. A triangular factorization of $k_2$ has the form

$$k_2=\left(\begin{matrix}1&\sum_{j=1}^{\infty}x_j^*z^{-j}\\0&1\end{matrix}\right)\left(\begin{matrix}\mathbf a_2&0\\0&\mathbf a_2^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_2(z)&\beta_2(z)\gamma_2(z)&\delta_2(z)\end{matrix}\right),$$

where $\mathbf a_2>0$ and the final factor has the required holomorphic or $VMO$ regularity. Random-phase triangular factorization conjecture. Under these assumptions, $k_2$ has a triangular factorization as in (II.3). The conjecture proposes a probabilistic condition that may ensure factorization despite the failure of $\zeta\in l^2$ alone to guarantee the needed cubic convolution convergence; the deterministic problem remains unresolved in the stated setting.