The maximal zero-set conjecture for sums of squares

For even $n$ and even degree $m$, let $P_{n,m}$ be the cone of non-negative real homogeneous forms in $n$ variables of degree $m$, and let $\Sigma_{n,m}$ be the subset consisting of sums of squares of real forms of degree $m/2$. Let $B'_{n,m}$ be the supremum of the number of isolated real zeros of a form in $\Sigma_{n,m}$. Maximal zero-set conjecture for sums of squares. For every pair $(n,m)$ with even $n$,

$$B'_{n,m}=\left(\frac n2\right)^{m-1}.$$

The proposed value is known when $m=3$, while the general even-dimensional case remains open.