On the suspension order of $(RP^{2m})^{[k]}$

It is shown that the suspension order of the $k$-fold cartesian product $(RP^{2m})^{[k]}$ of real projective $2m$-space $RP^{2m}$ is less than or equal to the suspension order of the $k$-fold symmetric product $SP^{k}RP^{2m}$ of $RP^{2m}$ and greater than or equal to $2^{r+s+1}$, where $k$ and $m$ satisfy $2^{r} \le 2m < 2^{r+1}$ and $2^{s}\le k<2^{s+1}$. In particular $RP^{2} \times RP^{2}$ has suspension order $8$, and for fixed $m\ge 1$ the suspension orders of the spaces $(RP^{2m})^{[k]}$ are unbounded while their stable suspension orders are constant and equal to $2^{\phi (2m)}$.