A counterexample for the Daugavet index of thickness in ${\mathrm{\ell }}_1$-sums

We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies $$T(X{\oplus }_{1}Y)=\min \{T(X),T(Y)\}\mathrm{.}$$ We show that this equality does hold whenever one of the two summands has the Daugavet property. On the other hand, if $D$ is a Banach space with the Daugavet property and $N$ is a suitable absolute norm, then for $X=D{\oplus }_{N}D$, one has $T(X{\oplus }_{1}X)<T(X)$.