Conjecture on finite expected condition numbers for random rank-one tuples

Let $n_1,n_2,n_3$ satisfy $n_1 \ge n_2 \ge n_3 \ge 3$, and let random rank-$1$ tuples of length $r$ in $\mathbb{R}^{n_1 \times n_2 \times n_3}$ have their expected condition number considered. Finite-expectation conjecture. There exists an integer

$$2 \le r^\star \le \frac{n_1 n_2 n_3}{n_1+n+2+n_3-2}$$

such that for all $1\leq r\leq r^\star$, the expected condition number is finite. The conjecture concerns the average computational difficulty and numerical sensitivity of CP decompositions; the paper provides empirical and impossibility evidence but does not establish the asserted existence of $r^\star$.