Let us say a sequence $(x\_n)\_{n=1}^\infty$ in some Banach space $X$ has $S\_C$ if there exist $k\_1<k\_2<\ldots$ such that for any $t\in \mathbb{N}$ and scalars $(a\_n)\_{n=1}^t$, $$\|\sum\_{n=1}^t a\_n x\_{k\_n}\|\leqslant C\Bigl(\sum\_{n=1}^t |a\_n|^2\Bigr)^{1/2}.$$ 


Let us say the Banach space $X$ has $HSP$ if every normalized, weakly null sequence in $X$ has $S\_C$ for some $C>0$. 


For $C>0$, let us say the Banach space $X$ has $C$-$HSP$ if every normalized, weakly null sequence in $X$ has $S\_C$. 


Is there a reflexive Banach space $X$ such that $X$ has $HSP$ but for each $C>0$, $X$ fails $C$-$HSP$? 



Non-uniform property of sequences
No, and the reflexivity plays no role. This is actually a theorem of [Knaust and Odell](http://helmut.knaust.info/paper/KOulp.pdf).


