How Deep Before the Signal Dies

A deep network is built entirely from sigmoid layers. At the current parameters the stack happens to be perfectly uniform: every layer carries the same weight $w = 2.5$, and every layer's sigmoid output is

$\sigma(z) = 0.8$

Backpropagation multiplies in one factor per layer as it travels downward. For a sigmoid layer that factor is

$d = w \cdot \sigma'(z)$

and the sigmoid's derivative is given by

$\sigma'(z) = \sigma(z)\bigl(1 - \sigma(z)\bigr)$

The loss hands the top of the stack a gradient of magnitude $G = 5.0$. After that gradient has passed down through $n$ of these layers, the magnitude reaching the weights below them is

$G \cdot d^{\,n}$

The team calls a layer dead once the gradient arriving at it has fallen strictly below $10^{-7}$ — at that size, multiplying by any sane learning rate moves the weights by nothing.

What is the smallest whole number $n$ at which the gradient has gone dead?

Give your answer as a whole number of layers.