High-probability convergence of the subspace power method

Let $\{a_i\}_{i=1}^R$ be $R$ independent standard Gaussian vectors in $\mathbb{R}^L$, let $c_n<\frac{1}{n!}$ be a constant, and consider the power method iterations given by the paper's iteration rule, with $x_1$ drawn uniformly from $\mathbb{S}^{L-1}$. High-probability convergence conjecture. If $R<c_nL^n$, then the power method iterations converge to $\pm a_j$ for some $j\in\{1,\dots,R\}$ with high probability, meaning with probability tending to $1$ as $L$ tends to infinity. The claim concerns the absence of bad local maxima in the corresponding optimization landscape and is motivated by numerical experiments; no proof or resolution is supplied in the source.