The question whether the states in $D=2m + 2$ dimensional string theory, which carry a representation of $SO(2m)$, span spaces which carry representations of $SO(2m+1)$ seems hopelessly complicated. For $m=1$, i.e in the most interesting case $D=4$, however, it boils down to the following question.

Let $h(x,q)$ be the function: $$ h(x,q) = \sum_N (q^N \sum_n h_{N,n} x^n) = \prod_{i=1}^{\infty} 1/((1-q^i*x)(1-q^i/x)) $$

Are the differences $d_{N,n}=h_{N,n}-h_{N,n+1}$ nonnegative for all $N\ge 2$ and all $n\ge 0$? ($h_{N,n}=0$ for $|n|>N$)

One easily sees $d_{N,N}=1, d_{N,N-1}=0$.

My calculations of about 20 nontrivial differences confirm the conjecture for $N\le 8$.

Added information: The numerical evaluation of $h(x,q)$ confirms the conjecture for $N\le 51$: http://www.itp.uni-hannover.de/~dragon/part1.erg , where the differences $d_{N,n}$ are listed as $A[N,n]$.

To exclude the special case $N=1$ one could add $q$ to $h(x,q)$ and try to prove that $$d(q,x)=(x-1/x)(q+h(x^2,q))=\sum_{m,n\ge 0}d_{m,n}q^m (x^{2n+1}-x^{-(2n+1)})$$ is a function with $d_{mn}\ge 0$ for all $m\ge 0$ and for all $n \ge 0\,$.