Conjecture on the decreasing quotient sequence of factorial minimality indices

Let $n_1,n_2,n_3,\ldots$ be the sequence defined by $\sigma_{n_\ell}=\ell+1$ for all $\ell\geq 1$, where the values of $\sigma_n$ arise from the paper's preceding construction. Quotient-sequence conjecture. The sequence

$$\frac{n_2}{n_1},\frac{n_3}{n_2},\frac{n_4}{n_3},\ldots$$

of quotients is strictly decreasing and

$$\lim_{i\longrightarrow\infty}\frac{n_{i+1}}{n_i}=e^2.$$

The observed initial quotients suggest this limiting pattern, while the preceding corollary only establishes the lower bound $n_{i+1}/n_i\geq 3$; the monotonicity and limiting value remain conjectural in the source.