To draw continued attention, here are some explanatory comments, with a tentative proof based on the idea of universal CSG.
The original quote (in my original post) is from wikipedia entry: Context-sensitive grammar, requote here: "There are even some context-sensitive grammars whose fixed grammar recognition problem is PSPACE-complete."
The statement that the word recognition problem for CSG is PSPACE-complete and the statement that the same kind of problem for some fixed grammar in CSG is PSPACE compelte are different! The former is sometimes called universal recognition problem for CSG, and the latter, well, may be called just fixed CSG grammar recognition problem. Let's refer to the two problems as UCSG and FCSG respectively.
The instances of UCSG are pairs of (G,x) where G can be any CSG, and x can be any strings; whereas those of FCSG for a fixed grammar G', are pairs of (G',x) where G' is fixed, and x can be any string.
Thus to say UCSG is PSPACE complete, it means that there is a polynomial time transformation T such that for any decision problem Q in PSPACE, there is pair (G,x)=T(Q) such that the answer to (G,x) is the answer to Q;
Whereas to say FCSG for a fixed CSG G' is PSPACE complete, it means that there is a polynomial time transformation T' such that for any decision problem Q in PSPACE, there is string x, such that (G',x)=T'(Q) is the equivalent problem to Q.
When the original author states that FCSG is PSPACE complete without giving a reference, he (she) may think that it is kind of obvious or its importance does not justify more detail. But I think a reference or certain hints are valuable here, assuming it is right.
Along this line of thinking, my tentative proof plan for FCSG being PSPACE complete is the following:
Let H be such a special grammar in CSG such that H accepts x if and only if x=u#v where u is the coding (by some coding funciton) of arbitrary grammar G(u) in CSG and G(u) accepts v. Since the condition of H is fulfillable by a LBA (or not so?), by the equivalence of LBA and CSG, such H exists; and any ICSG for such a H is PSPACE complete.
Since this H can simulate any other CSG, we may call it universal CSG.
Are there proofs along this line already, where, or this purported proof plan has flaws?
If the idea of universal CSG is essentially right, we may continue asking what other classes have their own universal instances.