Imagine a Schwarzschild black hole with mass $10^6 M_\odot$ in (almost) empty space. I remain at rest at a great distance and drop a robust clock. The time keeping mechanism of the clock is the radioactive decay of a suitable $\beta$ emitting isotope, insensitive to gravity. As the clock approaches the event horizon of the black hole, it "ticks" at an ever slower pace according to my wrist watch. I will never see the clock pass the even horizon, due to gravitational time dilation. So far, so good.
What about the clock at the instant it passes the event horizon in free fall? Does it tick just like it always did? I would say it ticks at its normal pace (although it's hard to imagine an experiment to verify this) because the event horizon isn't a special place for the clock. I've read quite a few related questions here, but some experts seem to disagree.
Answering the question "Equivalence principle and the meaning of the coordinate speed of light" John Duffield states
Even for the gedanken observer at the event horizon with his optical clock. Gravitational time dilation goes infinite. His clock is stopped, and he's stopped too. Contrary to what Kruskal-Szekeres coordinates suggest, the stopped observer doesn't see his stopped clock ticking normally "in his frame".
Duffield's answer appears in a strange grey on my monitor, and Javier disagrees. That's confusing for me.
My question: does the clock tick normally at the event horizon, according to an indestructible observer who also passes the event horizon in free fall, next to the clock?