No interior for sums of closures of the sets of multiple zeta-star values

For $p\geq 2$, let $ceta(\mathcal{T}_p)$ be the set associated with parameter $p$, let $coverline{\eta(\mathcal{T}_p)}$ denote its closure in $cmathbb{R}$, and let $a,b\in\mathbb{R}$.

No-interior conjecture. For every $p\geq 2$ and every $a,b\in\mathbb{R}$,

$$(a,b)\nsubseteq\overline{\eta(\mathcal{T}_p)}+\overline{\eta(\mathcal{T}_p)}.$$

Equivalently, $coverline{\eta(\mathcal{T}_p)}+\overline{\eta(\mathcal{T}_p)}$ has no interior points.

The conjecture is motivated by an analysis of Theorem 1.2 and by the Palis conjecture; the supplied text gives no resolution.