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These are my ITTM programs.

1 tape

$$\omega^{\omega+1}+1$$ steps

$$\gamma[10] ≥ \omega^{\omega+1}+3$$

* # * r *

0 _ 1 l 0
0 1 _ r 7
1 1 1 r 2
1 _ 1 l 5
2 1 _ r 1
2 _ 1 l 3
3 1 1 r 4
3 _ _ r 4
4 1 _ r 3
4 _ _ r 3
5 _ _ l 6
6 1 1 l 5
6 _ 1 l 0
7 _ _ r 1
8 _ 1 l 9
8 1 1 r halt
9 _ _ r 9
9 1 _ r 9

L _ _ r 7
L 1 _ r 8


3 tapes

$$\omega^{\omega2}+1$$ steps

$$\gamma[14] ≥ \omega^{\omega2}+71$$

* # * r *

0 _ _ r 1
0 1 _ r 1
1 _ _ r 2
1 1 _ r 2
2 _ 3 l 4
2 1 3 l 4
2 3 3 r 5
3 _ _ r 4
3 1 1 r 4
3 3 1 r 4
4 _ _ r 3
4 1 1 r 3
4 3 3 r 3
5 _ _ r 6
5 1 1 r 6
5 3 3 r 6
6 3 3 r 5
6 1 3 l 7
6 _ 1 l 9
7 _ _ l 8
7 1 1 l 8
7 3 3 l 8
8 3 1 l 7
8 _ _ r 1
9 _ _ l 10
9 1 1 l 10
9 3 3 l 10
10 3 1 l 9
10 _ 1 l 0
11 1 1 r halt
11 _ _ r 12
12 _ _ r 13
12 1 _ r 13
12 3 _ r 13
13 _ _ r 12
13 1 1 r 12
13 3 3 r 12
L _ _ r 1
L 1 3 r 11
L 3 _ r 0


$$\omega^{\omega2+3}$$ steps

$$\gamma[15] ≥ \omega^{\omega2+3}+72$$

* # * r *

0 _ _ r 1
0 1 _ r 1
1 _ _ r 2
1 1 _ r 2
2 _ 3 l 4
2 1 3 l 4
2 3 3 r 5
3 _ _ r 4
3 1 1 r 4
3 3 1 r 4
4 _ _ r 3
4 1 1 r 3
4 3 3 r 3
5 _ _ r 6
5 1 1 r 6
5 3 3 r 6
6 3 3 r 5
6 1 3 l 7
6 _ 1 l 9
7 _ _ l 8
7 1 1 l 8
7 3 3 l 8
8 3 1 l 7
8 1 7 l 14
8 _ _ r 1
9 _ _ l 10
9 1 1 l 10
9 3 3 l 10
10 3 1 l 9
10 _ 1 l 0
11 7 7 l 8
11 3 7 l 14
11 1 3 l 14
11 _ _ r 12
12 _ _ r 13
12 1 _ r 13
12 3 _ r 13
13 _ _ r 12
13 1 1 r 12
13 3 3 r 12
14 _ _ r 14
14 1 _ r 14
14 3 _ r 14
14 7 _ r 14
L _ _ r 1
L 1 3 r 11
L 7 _ r halt
L 3 _ r 0


Bounds

1 tape 3 tapes
1 $$\omega^2$$ $$≥\omega^4+4$$
2 $$≥\omega^3+1$$ $$≥\omega^7+8$$
3 $$≥\omega^4+2$$ $$≥\omega^{\omega}+12$$
4 $$≥\omega^5+3$$ $$≥\omega^{\omega+2}+20$$
5 $$≥\omega^6+4$$
6 $$≥\omega^7+5$$
7 $$≥\omega^\omega+1$$
10 $$≥ \omega^{\omega+1}+3$$
14 $$≥ \omega^{\omega2}+71$$
15 $$≥\omega^{\omega2+3}+72$$
22 $$≥ \omega^{\omega2}+5$$