Reflexive and strictly convex Banach spaces without Property (H)

We prove that Property $(H)$ fails for a broad class of reflexive strictly convex Banach spaces. More generally, the real Banach space \[ \Big(\bigoplus_{j=1}^{\infty} L^{p_j}(Ω_j,μ_j)\Big)_{\ell^s} \] does not have Property $(H)$ whenever $1\le s<\infty$, $1\le p_j<\infty$, $\sup_j p_j=+\infty$, an