What is the period of N xored identical random number generators, with differing seeds? This question is a kinda spin off from Why do we not combine random number generators?

If for example we had N identical type random number generators, with N different seeds, what would the period be if their outputs were xored together?  Would it be period, N x period or period (power N)?

So a numerical example would be:

2 linear feedback shift registers of 10 bits.

1 LFSR's period = 1023.

2 xored LFSRs' period = 1023, 2046 or 1046529?

(I have a sense that it's period but I can't overcome that fact that the state size is so much bigger.)
I will assume the N random generators have independent seeds and states.  Suppose their periods are $t_1,\dots,t_N$.  Then the period of their xor is a divisor of $\operatorname{lcm}(t_1,\dots,t_N)$.  Typically, it will be exactly $\operatorname{lcm}(t_1,\dots,t_N)$, but I don't know whether that is guaranteed in all cases.

If all $N$ generators have the same period $t$, then the period of their xor is a divisor of $t$ (and thus cannot be any larger than $t$).