Pfender's recurrence conjecture for critical clique edge densities

For each integer $k\ge 2$, let $d_k=d_k(K_k)$ denote the critical edge density of the complete graph $K_k$. Pfender's recurrence conjecture. The critical edge densities satisfy

$$d_2=0,\qquad d_k^2(1-d_{k-1})+d_k-1=0\quad\text{for }k\ge 3.$$

This conjecture concerns the still-undetermined critical edge densities $d_{\ell}(K_k)$ in cases not covered by the results mentioned in the source. The recurrence was proposed by Pfender for the case $\ell=k\ge 4$, and the source indicates that its validity was not known there.