The question is based on the proof of the main theorem of Gabai's paper on Foliations and the topology of 3-manifolds which is the following:

Theorem 5.1. Suppose $M$ is connected, and $(M,\gamma)$ has a sutured manifold hierarchy $$(M,\gamma) = (M_0,\gamma_0)\xrightarrow{S_1}(M_1,\gamma_1)\to\cdots\xrightarrow{S_n}(M_n,\gamma_n)$$ so that no component of $R(\gamma_i)$ is a compressing torus. Then there exist transversely oriented foliations $\mathcal{F}_0$ and $\mathcal{F}_1$ of $M$ such that the following hold.
$(1)$ $\mathcal{F}_0$ and $\mathcal{F}_1$ are tangent to $R(\gamma)$.
$(2)$ $\mathcal{F}_0$ and $\mathcal{F}_1$ are transverse to $\gamma$.
$(3)$ If $H_2(M,\gamma)\neq\emptyset$, then every leaf of $\mathcal{F}_0$ and $\mathcal{F}_1$ nontrivially intersects a transverse closed curve or a transverse arc with endpoints in $R(\gamma)$. However, if $\emptyset\neq\partial M = R_+(\gamma)$ or $R_-(\gamma)$, then this holds only for interior leaves.
$(4)$ There are no 2-dimensional Reeb components on $\mathcal{F}_i|\gamma,i=0,1$.
$(5)$ $\mathcal{F}_1$ is $C^\infty$ except possibly along Toral components of $R(\gamma)$ or $S_1$ if $\partial M = \emptyset$.
$(6)$ $\mathcal{F}_0$ is of finite depth.

The question is during the construction of $\mathcal{F}_1^{i-1}$ in case 2, i.e., $\partial T_i$ is contained in a component $V$ of $R(\gamma_i)$.
I'll follow the proof written in the paper: To construct $\mathcal{F}_1^{i-1}$ in this case, first glue $T_i^+$ and $T_i^-$ and produce a manifold $Q$ with an induced foliation $\mathcal{F}^1$ via $\mathcal{F}_1^i$. Let $f$ be the holonomy of the transverse annulus.
For $f = 1$ then following the construction of $\mathcal{F}_0^{i-1}$ get $C^\infty$ foliation. For $f\neq 1$ case, one can "push the holonomy to the boundary" to reduce the case to $f = 1$ case. The main part I'm not convinced is case 4:

If $V$ is a closed oriented surface of genus $>1$, and $f\neq 1$ then we need the following theorem.
Theorem (Mather-Sergeraert-Thurston). If $f:I\to I$ is a $C^\infty$ diffeomorphism satisfying $${d^n f\over dt^n}(\alpha) = \begin{cases} 1, & n=1,\\ 0, & n>1,\end{cases}$$ for $\alpha\in\{0,1\}$, then there exists $C^\infty$ diffeomorphisms $c_i,b_i:I\to I$, $i =1,2,\ldots,n$, satisfying the above conditions so that $$f\circ[c_1,b_1]\circ\cdots[c_n,b_n] = 1.$$ $\quad$Now let $Q_1$ be obtained by attaching thick bands $B_1$ and $C_1$ to $\partial Q$. Extend $\mathcal{F}^1$ to $\mathcal{F}^2$ on $Q_1$ by foliating these bands so that the holonomy along $B_1$ (or $C_1$) is $b_1$ (or $c_1$). One now observes that the holonomy along the new transverse annulus is $fc_1b_1c_1^{-1}b_1^{-1} = f\circ [c_1,b_1]$. By repeating this procedure $n$ times one gets a foliation $\mathcal{F}^{n+1}$ on $Q_n$ whose holonomy along the transverse annulus is trivial.

Question(s)
(1) In the last sentence, it says that repeating this procedure $n$ times. I understand the $n=1$ case by the drawing in the paper but I'm not convinced how the repetition can be done. How about the $n=2$ case or higher?
(2) It's a relatively minor question but I think the assumption of theorem 5.1 that *no component of $R(\gamma_i)$ is a compressing torus is just to avoid possible Reeb foliation? I don't understand why that assumption is required. It seems it's not specifically mentioned in the proof.

The question is based on the proof of the main theorem of Gabai's paper on Foliations and the topology of 3-manifolds which is the following:

Theorem 5.1. Suppose $M$ is connected, and $(M,\gamma)$ has a sutured manifold hierarchy $$(M,\gamma) = (M_0,\gamma_0)\xrightarrow{S_1}(M_1,\gamma_1)\to\cdots\xrightarrow{S_n}(M_n,\gamma_n)$$ so that no component of $R(\gamma_i)$ is a compressing torus. Then there exist transversely oriented foliations $\mathcal{F}_0$ and $\mathcal{F}_1$ of $M$ such that the following hold.
$(1)$ $\mathcal{F}_0$ and $\mathcal{F}_1$ are tangent to $R(\gamma)$.
$(2)$ $\mathcal{F}_0$ and $\mathcal{F}_1$ are transverse to $\gamma$.
$(3)$ If $H_2(M,\gamma)\neq\emptyset$, then every leaf of $\mathcal{F}_0$ and $\mathcal{F}_1$ nontrivially intersects a transverse closed curve or a transverse arc with endpoints in $R(\gamma)$. However, if $\emptyset\neq\partial M = R_+(\gamma)$ or $R_-(\gamma)$, then this holds only for interior leaves.
$(4)$ There are no 2-dimensional Reeb components on $\mathcal{F}_i|\gamma,i=0,1$.
$(5)$ $\mathcal{F}_1$ is $C^\infty$ except possibly along Toral components of $R(\gamma)$ or $S_1$ if $\partial M = \emptyset$.
$(6)$ $\mathcal{F}_0$ is of finite depth.

The question is during the construction of $\mathcal{F}_1^{i-1}$ in case 2, i.e., $\partial T_i$ is contained in a component $V$ of $R(\gamma_i)$.
I'll follow the proof written in the paper: To construct $\mathcal{F}_1^{i-1}$ in this case, first glue $T_i^+$ and $T_i^-$ and produce a manifold $Q$ with an induced foliation $\mathcal{F}^1$ via $\mathcal{F}_1^i$. Let $f$ be the holonomy of the transverse annulus.
For $f = 1$ then following the construction of $\mathcal{F}_0^{i-1}$ get $C^\infty$ foliation. For $f\neq 1$ case, one can "push the holonomy to the boundary" to reduce the case to $f = 1$ case. The main part I'm not convinced is case 4:

If $V$ is a closed oriented surface of genus $>1$, and $f\neq 1$ then we need the following theorem.
Theorem (Mather-Sergeraert-Thurston). If $f:I\to I$ is a $C^\infty$ diffeomorphism satisfying $${d^n f\over dt^n}(\alpha) = \begin{cases} 1, & n=1,\\ 0, & n>1,\end{cases}$$ for $\alpha\in\{0,1\}$, then there exists $C^\infty$ diffeomorphisms $c_i,b_i:I\to I$, $i =1,2,\ldots,n$, satisfying the above conditions so that $$f\circ[c_1,b_1]\circ\cdots[c_n,b_n] = 1.$$ $\quad$Now let $Q_1$ be obtained by attaching thick bands $B_1$ and $C_1$ to $\partial Q$. Extend $\mathcal{F}^1$ to $\mathcal{F}^2$ on $Q_1$ by foliating these bands so that the holonomy along $B_1$ (or $C_1$) is $b_1$ (or $c_1$). One now observes that the holonomy along the new transverse annulus is $fc_1b_1c_1^{-1}b_1^{-1} = f\circ [c_1,b_1]$. By repeating this procedure $n$ times one gets a foliation $\mathcal{F}^{n+1}$ on $Q_n$ whose holonomy along the transverse annulus is trivial.

Question(s)
(1) In the last sentence, it says that repeating this procedure $n$ times. I understand the $n=1$ case by the drawing in the paper but I'm not convinced how the repetition can be done. How about the $n=2$ case or higher?
(2) It's a relatively minor question but I think the assumption of theorem 5.1 that no component of $R(\gamma_i)$ is a compressing torus is just to avoid possible Reeb foliation? I don't understand why that assumption is required. It seems it's not specifically mentioned in the proof.