  Under $\text{CH}$ we construct a partition of Baire space into compact sets,
which is indestructible by countably supported iteration and product of Sacks
forcing of any length, answering a question of Newelski. Further, we present an
in-depth isomorphism-of-names argument for $\text{spec}(\mathfrak{a}_\text{T})
= \{\aleph_1, \mathfrak{c}\}$ in the product-Sacks model. Finally, we prove
that Shelah's ultrapower model for the consistency of $\mathfrak{d} <
\mathfrak{a}$ also satisfies $\mathfrak{a} = \mathfrak{a}_\text{T}$. Thus,
consistently $\aleph_1 < \mathfrak{d} < \mathfrak{a} = \mathfrak{a}_\text{T}$
holds relative to a measurable.
