Of course the grandparents like the six-year-old more. He actually exists outside someone else's body and has a personality and has bonded to them. It's sort of a stretch to think that they will dislike the upcoming one because of how much they like the current one.
From my perspective (from my own experience), it happened. And it's not that they actively disliked our child, it was just that he was the afterthought. If they thought of their grandchildren, they always thought of the other one first. We went on vacation with ILs once, and I had to endure the stories about the other grandkid until I'd just about bitten my lip off from not saying something impolite. They were WITH our kid, yet they couldn't even seem to enjoy him; they just had to keep talking about the other one. I can't tell you the number of times that we would be talking about something, and MIL would turn the conversation back to GC#1. We were supposed to visit them one time and DS was coming down with something (mild, but contagious). We called them to tell them, and the VERY first thing that MIL said after finding out was NOT to ask how DS was doing, but was to make a comment about how this would affect the other grandchild. Really?
And knowing my SIL and her entitlement tendencies and how the ILs seemed to think she can do no wrong, I predicted it. I remember telling my DH that SIL was going to have her parents babysit (i.e., claim their time) and they wouldn't have any time left for our kids. I didn't realize at the time how true this would be.
And yes, this is mostly on the grandparents, but I don't doubt that SIL cultivates this, as mine did.
And OP, not to be melodramatic, but if this 6 year old is that out of control, I would SERIOUSLY worry about him around a newborn. Even well-behaved children might feel displaced because of the attention lavished on a new baby, but I would worry that with his behavior and apparent lack of impulse control, he could do something harmful to the baby if he perceived it as a threat to his grandparents' attention. Remember, IL's house is his second home, so I'd imagine he'd feel pretty comfortable there and possessive of it.