Dilation Question for Veteran Moms

I had my cervix checked at 38+4 weeks and was told that it was high and closed. Got checked again at 39+1 weeks and still closed and baby has yet to drop at all. Now I got checked again today at 40+1 weeks and my cervix is still closed and high annd baby hasn't dropped at all. The doctor is talking about inducing next Tuesday. Has anyone else been through this? Was your cervix completely closed and not dilating this late and end up going into labor on your own?