Oh, yes, I understood that! I was more asking if you're 100% positive the egg came from that particular pen or if it's possible it could have been from another pen? Is an egg mix-up possible, in other words?
I've heard different estimations for how long hens can retain sperm, from only 10 days to upwards of 6 weeks. A month is pretty standard for clearing hens out, however. I would think that, while not necessarily
impossible, it would be pretty unlikely that she retained sperm from the other male in this case.

Have you seen the White rooster successfully breeding the hens?
So based on this, then, you are 100% positive that the egg is from the pen with the whites? If that is the case, then it's down to either retained sperm or one of your Whites is not recessive white. Maybe she's a very dilute Splash as was mentioned, maybe she's dominant white, or maybe she's some other combination of genes that makes unicolor white.
There
is a DNA test specifically for the recessive white gene, c, that Cochins generally have that you could try if you really wanted to get to the bottom of it without spending the time and energy on testcrossing... But it might be more worthwhile to grow out the chick and see if any other clues appear in its feathering later on before spending the money on that.
https://orders.iqbirdtesting.com/product/recessive-white-gene-c-in-chickens-with-blood-sample/
https://orders.iqbirdtesting.com/product/recessive-white-gene-c-in-chickens-with-eggshell-sample/
https://orders.iqbirdtesting.com/product/recessive-white-gene-c-in-chickens-with-feathers-sample/
Since recessive white, as Cochins typically are, could hide anything, it
is possible this rooster could have fathered such a chick...
Okay! Pictures can definitely be deceiving of such things, especially with white birds in my experience. I had to ask, though, because sometimes a small trait like that might go unnoticed, and that could indicate that there are other genes at play than what is expected of a pure Cochin.
I think, though, that you may have to let the chick grow and feather out a bit to see if there are any other clues to be had from that before we can give any definite answers on what's going on here.