Central limit conjecture for the Sudler product along Euler's number

Let $P_N(e)$ be the Sudler product associated with Euler's number $e$, whose continued fraction is $e=[2;1,2,1,1,4,1,\dots,1,2n,1,\dots]$. The relevant normalization suggested by the source is

$$\frac{\pi}{\sqrt{540}}\left(\frac{\log N}{\log\log N}\right)^{3/2}.$$

Euler Sudler-product central limit conjecture. For every $t\in\mathbb R$,

$$\frac{1}{M}\left|\left\{1\leq N\leq M:\frac{\log P_N(e)}{\frac{\pi}{\sqrt{540}}\left(\frac{\log N}{\log\log N}\right)^{3/2}}\leq t\right\}\right| \to\int_{-\infty}^{t}\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,\mathrm{d}x \quad\text{as }M\to\infty.$$

The conjecture is motivated by the Lindeberg-type condition for the continued fraction of $e$ and by the proved asymptotic variance of $\log P_N(e)$; it predicts Gaussian fluctuations around the negligible mean.