At the end of the talk I answered Hal's question wrongly.
Here is a correction:

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Hal's Q: What is the smallest graph with a disconnected strand?

Answer: I don't know what is the smallest example. 
The recipe for such example given in the talk requires many vertices, as we need a flag 2-sphere with 8 vertices s.t. any two of them are of distance at least 3. 

Note: the Stanley-Reisner ideal of the join of a 4-gon and a 6-gon does NOT have internal zeros in any strand. To see this, note that the join of two non-adjacent vertices from the 4-gon and a disconnected induced subcomplex of the 6-gon has a non-trivial first homology.