Conjecture on the large-length variance of left truncations

For a base $b\geq 2$ and a positive integer $\ell$, let $\mathrm{Trun}^L_b$ denote the number of left truncations and let $\mathrm{Var}_{b^\ell}(\mathrm{Trun}^L_b)$ be its variance over integers below $b^\ell$. Variance conjecture. As $\ell\to\infty$, we expect

$$\mathrm{Var}_{b^\ell}(\mathrm{Trun}^L_b)=\frac{1}{\log^2 b}\left(1-\frac{1}{b}\right)^2\left(\frac{b}{\phi(b)}-1\right)\log^2\ell+O_b(\log\ell).$$

The conjecture concerns the large-$\ell$ regime, where estimating the variance requires information about primes in arithmetic progressions with large moduli. The source presents heuristic evidence but no proof.