Centering Questions, and homeground Options

[Joker9125]

I was reading through the centering section of MiTS and I discovered that when centering for successes situational modifiers apply. For example

James a grade 1 Initiate is casting a force 3 fireball in an area of +1 background count. TN 4 + 1 (for background count) for a total TN of 5. He centers for successes using 6 die aganst TN of the spell plus modifiers or 5 in this case and gets 2 successes meaning he has 1 success to add to his sorcery test.

Correct so far? Please correct me if im wrong.

Now lets do the same test on James's homeground

James a grade 1 Initiate is casting a force 3 fireball in his homeground with a +1 background count. TN 4 - 1 (for homeground) + 1 (for background count) for a total TN of 4. He centers for successes using 6 die aganst the TN of the spell plus modifiers or 4 in this case. Now since centering is a seperate active skill would it get another -1 TN modifier? instead of a TN of 4 for the centering test would be a TN of 3?

For those that dont know the homeground edge lowers the TN of all active skill tests by 1 while on the homeground