Asymptotic Pfaffian-sum conjecture for general forms

Let $k$ be sufficiently large, and consider skew-symmetric $2k\times 2k$ matrices whose entries are linear forms in four variables. Their maximal pfaffians are forms of degree $k$.

Asymptotic Pfaffian-sum conjecture. A general form of degree $k$ can be expressed as a sum of $s$ pfaffians of such matrices, for

$$s=\frac{k}{24}+o(k).$$

This expectation comes from the dimension estimate for the variety of forms that are pfaffians of a single matrix and the corresponding secant-variety heuristic. It concerns the asymptotic sharpness of the general upper bound for expressing forms as sums of pfaffians; the source presents it as an expectation for $k\gg 0$.