Almost certainly not!
On the other hand, I have heard that a non-trivial part of the yield of a hydrogen bomb comes from the fast neutrons from the fusion causing a much more complete fissioning of the fissile material. Maybe, if the buried bomb was not damaged to the point where it was incapable of fusion ignition, the second bomb would contribute to the explosion in this way, without acting as a hydrogen bomb itself. With a high enough neutron flux from the first bomb, maybe the core of the second one would not have to undergo implosion, or even stay intact.
The article also talks about a 17-mile kill zone, and the creation of a new "North Carolina bay" despite the crash being 50 miles from Pamlico Sound. These seem to me to be incompatible claims, and surely the second, at least, must be hyperbole?
https://www.britannica.com/technology/nuclear-weapon/The-fir...
Richard Rhodes, "Dark Sun."
But that's in 1 atm! The initial wavefront of a fission device is going to be conservatively about 4 inches a microsecond (based on early above-ground test photos, modern high-yield devices would probably be faster still). This could very well turn the entire physics package of the secondary weapon into a thin pancake on the blast front, the plutonium can't get out of the way fast enough to avoid compression. This is a totally different scenario than a one-point-safety fizzle. The HE explosives in the second bomb might as well not exist relative to the overpressures faced from the first bomb's detonation.