This should be obvious but I'm not seeing it:

The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves coproducts and the localizing subcategory $\langle A\rangle$ of $\mathfrak T$ generated by $A$ is all of $\mathfrak T$.)

For instance, $\mathfrak T$ could be the derived category of a ring or a ring spectrum.

Let $\langle A\rangle_{\aleph_1}\subseteq\mathfrak T$ be the $\aleph_1$-localizing subcategory of $\mathfrak T$ (that is, the smallest triangulated subcategory containing $A$ and being closed under *countable* coproducts).

Certainly, if $B$ belongs to $\langle A\rangle_{\aleph_1}$, then $\mathfrak T_*(A,B)$ is a countably generated module over $\mathfrak T_*(A,A)$.

Question: Does the converse hold, that is, do we have $$ \langle A\rangle_{\aleph_1}=\{B\in\mathrm{Ob}(\mathfrak T)\mid\text{$\mathfrak T_*(A,B)$ is countably generated over $\mathfrak T_*(A,A)$}\}\;? $$