Show linear bounded automata accepting w is PSPACE-complete
Savitch's theorem shows that PSPACE=NPSPACE, which immediately implies that ALBA is in PSPACE.
In the other direction, suppose that $L$ is a language in PSPACE. Thus $L$ is accepted by some machine that uses space $p(n)$. We can construct another machine $L'$ which accepts an input $(x,y)$, erases $y$, and simulates $L$ on $x$. We can reduce $L$ to ALBA by mapping $L$ to $L'$ and $x$ to $(x,0^{p(|x|)})$.