Let $n_\lambda^K$  be the number all semi-standard Young tableaux of size 
$K$ with Ferrers diagrams diagram $\lambda$ 
(i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows 
and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds 
a representation of $GL(2m,C)$ ($m$ sufficiently large)  
and the sum $\sum_{\mu}N_{\lambda\mu} \mu$ of diagrams $\mu$, 
into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37). 
The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the
Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even 
subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even, $\delta_i\le \lambda_i$, 
$\delta_1\ge \delta_2\ge \dots$.
 

The conjecture states: 

For each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu $ combine
with multiplicities $l_\sigma^K\ge 0$ to families $\sigma$ of diagrams  which arise  upon 
restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

    $\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu =
    \sum_\sigma l_\sigma^K(\sum_{\bar\sigma}\bar\sigma)$

where for each $\sigma$ the family $(\sum_{\bar\sigma}\bar\sigma)$ consists of all diagrams $\bar\sigma$ with

    $\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Comment:

It is known, that at each level $K\ge 2$ the exited string states with vanishing momentum
span a vector space which allows a representation of $O(25)$.
This proves the conjecture to hold for all diagrams $\mu$ where the sum of the
lengths $\mu^\prime_1$ and $\mu^\prime_2$ of the first and second column does not exceed $24$.

If the conjecture can be shown to hold without the restriction $\mu^\prime_1 + \mu^\prime_2\le 24$
then the bosonic string is consistent in arbitrary dimension and the dimension 26 is just an artefact
of an unsuitable construction of the generators of Lorentz transformations in the lightcone
formulation.


[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991