Conjecture on the decreasing proportion of toric Fano varieties with exceptional collections

For each positive integer $d$, let $f(d)$ be the proportion of smooth projective toric Fano $d$-folds for which the Hanlon–Hicks–Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles. The known values are

$$\frac{1}{1}=f(1)=f(2)=\frac{5}{5},\qquad f(3)=\frac{16}{18},\qquad f(4)=\frac{72}{124},\qquad f(5)=\frac{300}{866}.$$

Monotonicity and vanishing conjecture. The function $f(d)$ is strictly monotone decreasing for $d>1$, and

$$\lim_{d\rightarrow\infty} f(d)=0.$$

This conjecture extrapolates from the computed proportions in dimensions one through five. It predicts that the proportion of smooth projective toric Fano varieties detected by this resolution as having such exceptional collections decreases strictly in higher dimensions and tends to zero; no resolution of the conjecture is stated.