Would northerners accept a Lannister child by Sansa to inherit the North?

Would northerners accept a Lannister child by Sansa to inherit the North? He would be Ned Stark’s grandson. Would it depend on raising him in Winterfell, in the Northern way? What does Taiwin expect?
it wouldn't have ever gotten to that point.
On the one hand, yes, Ned’s line is respected and that means something. On the other, the Lannisters killed Ned and it’d be pretty obvious that Sansa was an unwilling participant in all this and that, going by Taiwin’s probable plan - she wasn’t actually in charge as Lady of Winterfell. Not to mention the fact that Ned’s sons, who came before Sansa, would also have (as far as the northerners know) been killed by treachery.
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But: It’d likely be settled before Sansa had a Lannister son who would ever be in any position to rule. If nothing else, it looks quite likely that Robb disinherited her in his will (which can always be set aside if she turns up under her own power), essentially giving the northmen permission to pass her (and Tyrion) over in favor of, in all likelihood, Jon, or if not him, then someone else, whoever is next in line like the Royce branch or another northern family that might win a Kingsmoot-type contest. Jon wouldn’t, from what he tells Stannis,, want to bypass Sansa, but I think even he would if he thought or knew she was under duress and that there was a risk of the Lannisters oppressing the North or abusing their authority through her. And of course, unbeknownst to everyone, almost, both Bran and Rickon are still alive and either one would trump Sansa. We know the Manderlys at least know where Rickon is and might even have been keeping tabs on him, they’re not keeping his survival a secret if the alternative is Tyrion being Lord of Winterfell.
Taiwin seems to have thought the northmen were or would be cowed and that the Boltons could do the hard work of pounding them into submission before being unceremoniously swept aside for Taiwin to rule through Tyrion through Sansa. But Taiwin needed to rely on the lowest of low treachery to defeat Robb, he couldn’t do it in the field, and the Lannister army. what Tywin would need to enforce his will, is unlikely to have much success that far north, especially as the seasons shift closer to winter.
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I think if the rest of the Starks where dead and the Northerners knew this Lannister x Stark child and it grew up in Winterfell they would be more likely to accept it, especially if Jon Snow accepted its father and told the rest of the Northerners that he wasn’t like the other Lannisters which he probably would if he were alive.