Transformation of the answer of Nik, for an hyperfinite infinite dimensional intersection:

He gave the counterexample $(\mathcal{A} \otimes I_2)' \cap (\mathcal{B} \otimes I_2)' \simeq M_2(\mathbb{C})$, for $\mathcal{A}$, $\mathcal{A'}$, $\mathcal{B}$, $\mathcal{B'}$ ${\rm II}_1$-factors and $\mathcal{A} \cap \mathcal{B} = \mathcal{A'} \cap \mathcal{B'} = \mathbb{C}$.
Let two states on $(\mathcal{A} \otimes I_2)^{\otimes \infty}$ and $(\mathcal{B} \otimes I_2)^{\otimes \infty}$ generating ${\rm II}_1$-factors $\mathcal{M}=\overline{(\mathcal{A} \otimes I_2)^{\otimes \infty}}$ and $\mathcal{N}=\overline{(\mathcal{B} \otimes I_2)^{\otimes \infty}}$ such that $\mathcal{M'}=\overline{(\mathcal{A'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ and $\mathcal{N'}=\overline{(\mathcal{B'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ are ${\rm II}_1$ factors, $\mathcal{M} \cap \mathcal{N} = \mathbb{C} $ and $\mathcal{M'} \cap \mathcal{N'} \simeq \overline{M_2(\mathbb{C})^{\otimes \infty}} \simeq \mathcal{R}$ the hyperfinite ${\rm II}_1$-factor.
For more details about infinite tensor product of von Neumann algebras, see this answer of Nik.

Generalized question (posted here):
Let $\mathcal{A}_1 \dots \mathcal{A}_n \subset B(H)$ be ${\rm II}_1$-factors such that $\forall i \, \, \mathcal{A}_i'$ is also a ${\rm II}_1$-factor and $\bigcap_i \mathcal{A}_i = \mathbb{C}$.
Is it true that $\bigcap_i \mathcal{A}'_i$ is hyperfinite? Else, what are counterexamples?

Transformation of the answer of Nik, for an hyperfinite infinite dimensional intersection:

He gave the counterexample $(\mathcal{A} \otimes I_2)' \cap (\mathcal{B} \otimes I_2)' \simeq M_2(\mathbb{C})$, for $\mathcal{A}$, $\mathcal{A'}$, $\mathcal{B}$, $\mathcal{B'}$ ${\rm II}_1$-factors and $\mathcal{A} \cap \mathcal{B} = \mathcal{A'} \cap \mathcal{B'} = \mathbb{C}$.
Let two states on $(\mathcal{A} \otimes I_2)^{\otimes \infty}$ and $(\mathcal{B} \otimes I_2)^{\otimes \infty}$ generating ${\rm II}_1$-factors $\mathcal{M}=\overline{(\mathcal{A} \otimes I_2)^{\otimes \infty}}$ and $\mathcal{N}=\overline{(\mathcal{B} \otimes I_2)^{\otimes \infty}}$ such that $\mathcal{M'}=\overline{(\mathcal{A'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ and $\mathcal{N'}=\overline{(\mathcal{B'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ are ${\rm II}_1$ factors, $\mathcal{M} \cap \mathcal{N} = \mathbb{C} $ and $\mathcal{M'} \cap \mathcal{N'} \simeq \overline{M_2(\mathbb{C})^{\otimes \infty}} \simeq \mathcal{R}$ the hyperfinite ${\rm II}_1$-factor.
For more details about infinite tensor product of von Neumann algebras, see this answer of Nik.

Generalized question (posted here):
Let $\mathcal{A}_1 \dots \mathcal{A}_n \subset B(H)$ be ${\rm II}_1$-factors such that $\forall i \, \, \mathcal{A}_i'$ is also a ${\rm II}_1$-factor and $\bigcap_i \mathcal{A}_i = \mathbb{C}$.
Is it true that $\bigcap_i \mathcal{A}'_i$ is hyperfinite? Else, what are counterexamples?

After the answer of Nik, giving the counterexample $(\mathcal{A} \otimes I_2)' \cap (\mathcal{B} \otimes I_2)' \simeq M_2(\mathbb{C})$, for $\mathcal{A}$, $\mathcal{A'}$, $\mathcal{B}$, $\mathcal{B'}$ ${\rm II}_1$-factors and $\mathcal{A} \cap \mathcal{B} = \mathcal{A'} \cap \mathcal{B'} = \mathbb{C}$, it appears that there are also counterexamples with hyperfinite infinite dimensional intersection, by putting a state on $(\mathcal{A} \otimes I_2)^{\otimes \infty}$ and on $(\mathcal{B} \otimes I_2)^{\otimes \infty}$ generating ${\rm II}_1$ factors $\mathcal{M}=\overline{(\mathcal{A} \otimes I_2)^{\otimes \infty}}$ and $\mathcal{N}=\overline{(\mathcal{B} \otimes I_2)^{\otimes \infty}}$ such that $\mathcal{M'}=\overline{(\mathcal{A'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ and $\mathcal{N'}=\overline{(\mathcal{B'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ are ${\rm II}_1$ factors, $\mathcal{M} \cap \mathcal{N} = \mathbb{C} $ and $\mathcal{M'} \cap \mathcal{N'} \simeq \overline{M_2(\mathbb{C})^{\otimes \infty}} \simeq \mathcal{R}$ the hyperfinite ${\rm II}_1$-factor.
For more details about infinite tensor product of von Neumann algebras, see this answer of Nik.

Generalized question (posted here):
Let $\mathcal{A}_1 \dots \mathcal{A}_n \subset B(H)$ be ${\rm II}_1$-factors such that $\forall i \, \, \mathcal{A}_i'$ is also a ${\rm II}_1$-factor and $\bigcap_i \mathcal{A}_i = \mathbb{C}$.
Is it true that $\bigcap_i \mathcal{A}'_i$ is hyperfinite? Else, what are counterexamples?

Transformation of the answer of Nik, for an hyperfinite infinite dimensional intersection:

He gave the counterexample $(\mathcal{A} \otimes I_2)' \cap (\mathcal{B} \otimes I_2)' \simeq M_2(\mathbb{C})$, for $\mathcal{A}$, $\mathcal{A'}$, $\mathcal{B}$, $\mathcal{B'}$ ${\rm II}_1$-factors and $\mathcal{A} \cap \mathcal{B} = \mathcal{A'} \cap \mathcal{B'} = \mathbb{C}$.
Let two states on $(\mathcal{A} \otimes I_2)^{\otimes \infty}$ and $(\mathcal{B} \otimes I_2)^{\otimes \infty}$ generating ${\rm II}_1$-factors $\mathcal{M}=\overline{(\mathcal{A} \otimes I_2)^{\otimes \infty}}$ and $\mathcal{N}=\overline{(\mathcal{B} \otimes I_2)^{\otimes \infty}}$ such that $\mathcal{M'}=\overline{(\mathcal{A'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ and $\mathcal{N'}=\overline{(\mathcal{B'} \otimes M_2(\mathbb{C}))^{\otimes \infty}}$ are ${\rm II}_1$ factors, $\mathcal{M} \cap \mathcal{N} = \mathbb{C} $ and $\mathcal{M'} \cap \mathcal{N'} \simeq \overline{M_2(\mathbb{C})^{\otimes \infty}} \simeq \mathcal{R}$ the hyperfinite ${\rm II}_1$-factor.
For more details about infinite tensor product of von Neumann algebras, see this answer of Nik.

Generalized question (posted here):
Let $\mathcal{A}_1 \dots \mathcal{A}_n \subset B(H)$ be ${\rm II}_1$-factors such that $\forall i \, \, \mathcal{A}_i'$ is also a ${\rm II}_1$-factor and $\bigcap_i \mathcal{A}_i = \mathbb{C}$.
Is it true that $\bigcap_i \mathcal{A}'_i$ is hyperfinite? Else, what are counterexamples?