Ottaviani–Shapiro conjecture on isolated zeros of sums of squares

Let $\widetilde\sharp(2k,l)$ denote the maximal possible number of isolated zeros of real non-negative polynomials of degree $2k$ in $l$ variables that are representable as sums of squares of real polynomials of degree at most $k$.

Ottaviani–Shapiro conjecture. For any number of variables,

$$\widetilde\sharp(2k,l)=k^l.$$

The equality is known in the special case $l=2$, while the assertion for arbitrary numbers of variables is open.