It could lose its world heritage status if the proposed tunnel is constructed in its current form.
All it does is tunnel under the existing A303 for 1.8km directly opposite the henge. Its entrances and exits are ultra modern and physically alter the landscape of what is already a Site of Special Scientific Interest (SSSI). The road is already there so there is not much that can be done about it, but to improve its function by ruining a unique landscape seems silly in the extreme.
I do not know why Highways England do not bite the bullet and build a tunnel from the end of the Amesbury bypass to the junction of the A360. It could be built deep enough to leave the upper layers intact for further archeology and if it was a straight line it would be further away from the henge.
£s billions can be thrown at tunnels under London so Crossrail can exist, but we penny-pinch on a unique, one-off, prehistoric monument some 5,000 years old. Its construction is so special that scientists and engineers still cannot figure it out. They have tried replicating some aspects using modern day knowledge and materials and none of them have been a roaring success.
Leave the place in peace.