Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two preimages of $f_i$ under $\operatorname{NS}(X) \twoheadrightarrow \operatorname{Num}(X)$ inducing the usual ten elliptic pencils $|2F_1^\pm|, \dots, |2F_{10}^\pm|$ on $X$.
By taking $L_i = \mathcal{O}_X(F_i^+)$ (initial choice of $\pm$ doesn't matter since they can be switched by tensoring with $K_X$ without affecting orthogonality, though this does give rise to $2^{10}$ different exceptional collections), it is well-known (see [1],[2]) that
$$ \mathscr{L} := \{L_1, \dots, L_{10}\} $$
is a completely orthogonal exceptional collection in $D^\flat(X)$. As pointed out in [1, §5], the admissible subcategory $\operatorname{Ku}(X, \mathscr{L}) = \langle L_1, \dots, L_{10}\rangle^\perp $ (sometimes referred to in literature as the Kuznetsov component) should have a rank 2 Grothendieck group.
What I am trying to figure out is how to find two independent generators of $K_0( \operatorname{Ku}(X, \mathscr{L}) )$ from our isotropic 10-sequence. After looking at [2], it seems the 3-spherical objects $S_i$ obtained from the cocones of the Serre functor
$$ S_i \to L_i \to S_{D^\flat(X)}(L_i) $$
are the interesting objects in our 3-CY category $\operatorname{Ku}(X, \mathscr{L})$; the main issue is that these objects are easily seen to be numerically trivial with $\chi(S_i, S_j) = 0$ for all $1 \leq i,j \leq 10$. How would one go about finding generators based on $\mathscr{L}$ in this case in order to completely compute the Euler pairing $\chi(-, -)$ on $D^\flat(X)$?
References:
[1] Zube, S. Exceptional vector bundles on Enriques surfaces. Math Notes 61, 693–699 (1997). https://doi.org/10.1007/BF02361211
[2] C. Li, H. Nuer, P. Stellari, X. Zhao. A refined derived Torelli theorem for Enriques surfaces. Math. Ann. 379 (2020), 1475–1505.
