The conjecture on asymptotic normality and log-normality of hidden-word counts

Let $Z$ be the number of occurrences of a pattern $w$ as a subsequence in a random text of length $n$, and let $\binom{n}{m}$ be the number of possible positions for a subsequence when $w$ has length $m$. Let $\tfrac{\sigma_1^2}{\binom{n}{m}^2}$ denote the variance scale appearing in the first-order projection. Asymptotic normality and log-normality conjecture. If

$$\sigma_1^2=o\bigl(\binom{n}{m}^2\bigr),$$

or equivalently

$$\sigma_1^2=o\bigl(\frac{n^2}{m^2}\binom{n-1}{m-1}^2\bigr),$$

then

$$Z/\operatorname{\mathbb E{}}Z\sim \operatorname{AsN}\Bigl(1,\frac{\sigma_1^2}{\binom{n}{m}^2}\Bigr).$$

Moreover, at least as long as $m=o(n)$,

$$\ln Z\sim \operatorname{AsN}\Bigl(a_n,\frac{\sigma_1^2}{\binom{n}{m}^2}\Bigr)$$

for some sequence $a_n$. The conjecture extends the asymptotic normality results proved earlier in the paper and predicts a corresponding log-normal regime under the stated variance condition.