Route 1

Does anybody know if they started to do construction on the upgraded route 1?

Between Beit Shemesh and Jerusalem? No. I was there two days ago.
My impression is they are not immediately about to start construction. (And if you ask me, they would be stupid to start before the fast train to Jerusalem is finished in a few years.)

However, they are really tearing up a wide path just to the east of road 38 north of Beit Shemesh. While now it seems they are laying pipes there, I assume that is the prelude to the construction of 2 more lanes for road 38.