Will this recruit look to sign at the very beginning of recruiting phase 2?

Lot of factors. They can sign on that first cycle. But not necessarily. If you're in a battle for him, it could be the very last cycle of the very last day. It's safe to say he won't likely sign in RS1. It mostly comes down to the battle going on for the recruit.

If you're very high, and no one else is above very low, Its possible you could sign him on that very first cycle. I just had this exact thing happen 2 days ago.

I tend to look at signing preferences as a general time frame and nothing definite. Because of the many factors involved. Like who's competing for him, what division those competing coaches are in, and what division the recruit is projected at.

Here's how signings work.

At the beginning of the cycle, the recruit will determine if it 'wants' to sign or not. It will first look at it's signing preference then qualifying offers.

Let's say he's a late signer. He will always choose to NOT sign in the 1st session. At the very first cycle of the 2nd session, it will run the RNG on whether it 'wants' to sign. I'm not sure of the exact starting % of the RNG but let's call it 10%. That guy will say "I will sign this period" then it looks to see if any team qualifies (above Moderate and scholly offer). If so then he signs.

If the RNG of signing decision says he doesn't want to sign or there is no qualifying offer, he doesn't sign and waits until the next cycle. At the start of the next cycle, the 'wants to sign' RNG % increases (I'm not sure the increase % but let's say it's another 10%).

The process just keeps repeating itself with ever increasing odds he will sign.

Short answer

There is a very small chance a Late signer preference will sign the very 1st cycle of the 2nd session. It's possible but not probable.