Answer One (mostly) to that is in two pieces, located in two books. One:
Here was the unborn jihad, he knew. Here was the race consciousness
that he had known once as his own terrible purpose. Here was reason
enough for a Kwisatz Haderach or a Lisan al-Gaib or even the halting
schemes of the Bene Gesserit. The race of humans had felt its own
dormancy, sensed itself grown stale and knew now only the need to
experience turmoil in which the genes would mingle and the strong new
mixtures survive. All humans were alive as an unconscious single
organism in this moment, experiencing a kind of sexual heat that could
override any barrier.
And Paul saw how futile were any efforts of his
to change any smallest bit of this. He had thought to oppose the jihad
within himself, but the jihad would be. His legions would rage out
from Arrakis even without him. They needed only the legend he already
had become. He had shown them the way, given them mastery even over
the Guild which must have the spice to exist.
in Dune, and the other:
The life of the prophet locks us into his vision, Leto thought. And a
prophet could only break out of the vision by creating his death at
variance with that vision. That was how it appeared in Leto’s doubled
vision, and he pondered this as it related to the choice he had made.
Poor Baptist John, he thought. If he’d only had the courage to die
some other way … But perhaps his choice had been the bravest one. How
do I know what alternatives faced him? I know what alternatives faced
my father, though.
Leto sighed. To turn his back on his father was
like betraying a god. But the Atreides Empire needed shaking up. It
had fallen into the worst of Paul’s vision. How casually it
obliterated men. It was done without a second thought. The mainspring
of a religious insanity had been wound tight and left ticking.
And we’re locked in my father’s vision
A way out of that insanity lay
along the Golden Path, Leto knew. His father had seen it. But humanity
might come out of that Golden Path and look back down it at Muad’Dib’s
time, seeing that as a better age. Humankind had to experience the
alternative to Muad’Dib, though, or never understand its own myths.
Security … peace … prosperity …
Given the choice, there was little
doubt what most citizens of this Empire would select
in Children of Dune.
My understanding is that Paul Atreides' failure at avoiding The Golden Path, but still unleashing "Muad'Dib's Fanatics" on the universe was the spark that would ignite ultimate religious war.
Answer Two - after some hard thinking - is probably buried in the two quotes from God Emperor of Dune:
“I point out to you, Marcus Claire Luyseyal, a lesson from past over-machined societies which you appear not to have learned. The devices themselves condition the users to employ each other the way they employ machines.”
And
She nodded gravely, prompting: “At one time…”
“The lxians contemplated making a weapon-a type of hunter-seeker, self-propelled death with a machine mind. It was to be designed as a self improving thing which would seek out life and reduce that life to its inorganic matter.”
“I have not heard of this thing, Lord.”
“I know that. The lxians do not recognize that machine makers always run the risk of becoming totally machine. This is ultimate sterility. Machines always fail … given time. And when these machines failed there would be nothing left, no life at all.
I believe the second... option is not the right answer, if we treat the quote from OP literally (humanity, extinct). However, it may be good to put it in here, to provide context for my original answer, now Option One. Because it is provides for the possibility of it happening during Leto's reign, while still guiding humanity onto The Golden Path. Because it also means it could happen by the time it's mentioned.
I also believe that the quote provided in the comments (from Siona's vision from the Test) is not the answer. Machines -in general - would be the demise of the humans EVENTUALLY, but not in the timeframe of the OP (between initiating The Golden Path and the "now" of the question).
