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 $C$-$HSP$?