I'm wondering under what hypothesis it is true a property like

$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$

where $\mathcal{H}_2\hookrightarrow \mathcal{H}_1$ are Hilbert spaces contained in a larger Hilbert space $\mathcal{H}$ with $X\subset \mathcal{H}$.

I'm not skilled in interpolation theory, but here is my attemp. In Triebel's book Section 1.17.1 (https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/18) there is a Theorem which read as follows

Theorem 1: Let $\{A_0, A_1\}$ be an interpolation couple. Let $B$ be a complemented subspace of $A_0+A_1$ whose projection belongs to $L(\{A_0, A_1\}, \{A_0, A_1\})$. Let $F$ be an arbitrary interpolation functor. Then $\{A_0\cap B, A_1\cap B\}$ is also an interpolation couple and $$F(\{A_0\cap B, A_1\cap B\})=F(\{A_0, A_1\})\cap B$$

In my case, the interpolation couple would be $\{\mathcal{H}_2, \mathcal{H}_1\}$ and $B=\mathcal{H}_1\cap X$. If $X$ is such that $\mathcal{H}_1\cap X$ is a closed subspace of $\mathcal{H_1}$ then it is also a complemented subspace of $\mathcal{H}_1$ whose projection is linear continuous in $\mathcal{H}_1$ (i.e. belongs to $L(\mathcal{H}_1)$). The previous reasoning implies that I'm able to apply the previous Theorem to arrive my initial statement, or I'm missing something?

I know that interpolation is not well behaved with respect to restriction (https://math.stackexchange.com/questions/3542640/complex-interpolation-and-intersection) and I didn't find much more results than the previous one in the literature. Every hint or reference is very well received!

Remark: I asked in some generality, but I'm treating a particular case where $\mathcal{H_2}, \mathcal{H_1}$ are sobolev spaces $H^k$, the larger space is $L^2$ and $X$ is the domain of a maximal monotone operator, in some interval $(0, L)$.