I would like to ask a question about phantom categories and strongly exceptional collections.

Let $X$ be a smooth prpjective variety over an algebraically closed field. An admissible subcategory $B \subset D^{b}(X)$ is said to be a **phantom category** if $K(B) = 0$ but $B \neq 0$ (where $K(B)$ is the K-theory of $B$).

I know (for instance from here : https://arxiv.org/pdf/1209.6183.pdf) that there exists smooth projective varieties having an exceptional collection of vector bundles $E_1, \ldots, E_n$ such that $n = \textrm{rank}(K(X))$, but the collection is not full. This implies in particular that the left orthogonal to the category generated by the $E_i$ is a phantom category.

I was wondering if there exists examples as above, but with the additional assumption that the collection $E_1,\ldots,E_n$ is **strongly exceptional**?
(that is the higher $\mathrm{Ext}$ between the $E_i$'s vanish in both directions).

In all examples I have been reading so far, the exceptional collections of length equal to $\mathrm{rank}(K(X))$ which are not full are **never** strongly exceptional. Does someone know an example where this could be the case?

Thanks in advance for your answers!