What is tight NSPACE complexity of $ALT\text{-}SPACE(a(n),s(n))$?
It's open, at least for the inclusion 
$ALTSPACE(\alpha(n),\log n)\subseteq NSPACE(f(n))$ with $\Omega(\log n)\leq f(n)\leq \alpha(n)^{o(1)}$. According to Chandra, Kozen, and Stockmeyer, we know
$$DTIME(n^k)\subseteq ALTSPACE(n^{2k},\log n)\subseteq \mathsf{P}.$$

Therefore if it is true, then $\mathsf{P}\neq \mathsf{NPSPACE}$ which is not known to be true. On the other hand, rejecting it will yield $\mathsf{P}\neq \mathsf{NL}$, which is also not known to hold.