I doubt it. The White House is officially on record as supporting the House Bill.
They will just do what they have always done. Ignore the parts of the law that they don't want to enforce.
Bingo. Nail on head.
Tie the number of H1B visas to demonstrated enfocement guidelines. e.g. 10,000 additional H1B visas when the estimated number of illegals in the country drops below 10 million. 10,000 additional H1Bs and 100,000 additional immigrant visas for each million it drops thereafter. A reasonably high bar for immigrant visas such as a demonstrated history of law-abidy productivity, tax payment, ability to communicate in English, military service, etc. Enough positives could even overcome illegal entry-- but for illegal entrants without enough positives, they would have to get in line behind those outside of the country wanting in.