The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.

An operator $T:X\to Y$ is $p$-convergent if it takes weakly $p$-summable sequences to norm null ones.

A Banach space $X\in C_p$ if the identity map $i$ on $X$ is $p$-convergent.

It is known that $X\in C_p$ if and only if every operator $T:\ell_{p^*}\to X$ is compact, where $p^*$ is the conjugate of $p$.

Is $Y\in C_p$ for some $1<p<\infty$ ? Is $Y\in C_2$?