A distributional convergence conjecture for DCPD deflation

Let $\mathscr{T}$ be the tensor space and define

$$\mathscr{T}^{(R)}=\{\boldsymbol{\mathcal{T}}\in\mathscr{T}:\operatorname{rank}\{\boldsymbol{\mathcal{T}}\}\leq R\}.$$

For $L>1$, let $Z_l=\|\boldsymbol{\mathcal{E}}[R,l]\|$ and define

$$F_L[\beta]=P\left(Z_L\leq\sin(\beta)Z_{L-1}\leq\cdots\leq(\sin(\beta))^{L-1}Z_1\right).$$

DCPD distributional convergence conjecture. There exists at least one absolutely continuous probability measure $\mu$ on tensors $\boldsymbol{\mathcal{T}}$ in $\mathscr{T}^{(R)}$ such that (i) for every $\varepsilon\in(0,1]$ and every $l>1$, there exists $\beta\in[0,\pi/2)$ with $F_l[\beta]>1-\varepsilon$, and (ii) for every $l>1$, there exists $\beta\in[0,\pi/2)$ such that $(\sin(\beta))^{l-1}$ is a strictly monotonically decreasing sequence converging to $0$. These conditions are intended to ensure that the residual norms tend to zero and that DCPD converges to an exact CP decomposition. The supplied text gives no resolution status beyond stating the conjecture.