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 $GL(2m,C)$
and the sum $\sum_{\mu:l(\mu)\le 2m}N_{\lambda\mu} \mu$ of diagrams $\mu$,
into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).

Here $l(\mu)=\mu_1^\prime + \mu_2^\prime$ denotes the sum of the lengths
of the first and the second column of $\mu$.

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, $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

Then it is known from string theory, that for $m=12$ and for each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu:l(\mu) \le 2m} n_\lambda^K N_{\lambda\mu} \mu $ combines with multiplicities $l_\sigma^{K,m}\ge 0$ 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)

$$\sum_{\lambda\mu:l(\mu)\le 2m} n_\lambda^K N_{\lambda\mu} \mu = \sum_\sigma l_\sigma^{K,m}(\sum_{\bar\sigma in \sigma}\bar\sigma)$$

where $\bar \sigma$ is in the family $\sigma$ if

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

Does this relation hold for other $m$? Can one show that its validity for $m$ implies it for smaller $m^\prime$?

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