I'm not sure the watermark has been demonstrated to have that kind of property. Even if you use a CSRNG as the 'key' once you're feeding it through the token selection process it's going to risk re-introducing certain correlations.
To give an example in the extreme: if you make the bias total and only make that decision based on the preceding token, there are certain token pairs that your model will never output, and this will be pretty obvious even to people just reading the text (because of any given common two-token phrase, there's a 50% chance you would just disappear in watermarked text). You can make this less extreme and more hidden by increasing the window and reducing the bias, but at the cost of reducing the signal. I don't know exactly what the tradeoff curve looks like, so it might be that you can reach set of parameters where the bias is in principle undetectable without the key but still reliably detectable for realistic lengths of text segments, but I would not assume that this is definitely the case.