Recommended usage with random seeds and streams

[LTLA]

First off: apologies if this is the wrong place to ask a usage-related question, I couldn't find any indication of where to ask for help on the PCG website.

The scenario is fairly simple: I am using multiple pcg32 instances (at least 1000, up to a million), each one of which is constructed with a different seed and an incremented stream value (1, 2, 3, etc.). The seeds are different between instances; for the sake of discussion, they can be assumed to be independently and uniformly sampled from the entire range of 64-bit integers.

Upon reviewing my code, someone mentioned that the use of different seeds was unnecessary in the presence of different streams. This sounded reasonable but some reading of the PCG blog made me wonder:

it may be easier than I had thought to make “nearby” streams with correlated initializations like a constant seed and streams of 1,2,3,4

So, the question: is there any advantage or disadvantage to my current PCG32 initialization scheme, over using a constant seed and a different stream for each instance?