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 $$(\underset{j=1}{\overset{\mathrm{\infty }}{⨁}}{L}^{p_j}({\mathrm{\Omega }}_j,{\mu }_j){)}_{{\mathrm{\ell }}^s}$$ does not have Property $(H)$ whenever $1\le s<\mathrm{\infty }$, $1\le p_j<\mathrm{\infty }$, ${\sup }_j{p}_j=+\mathrm{\infty }$, and each $L^{p_j}({\mathrm{\Omega }}_j,{\mu }_j)$ is infinite-dimensional.