Doeraene–El Haouari conjecture on sectional and relative category

Let $f:Y\rightarrow X$ be a map. A homotopy retraction of $f$ is a map $r:X\rightarrow Y$ such that $rf\simeq \operatorname{id}_Y$. The sectional category of $f$, denoted by $\operatorname{secat}(f)$, and its relative category, denoted by $\operatorname{relcat}(f)$, are the invariants defined using the iterated join $j_f^n:*_X^nY\rightarrow X$; in particular, $\operatorname{secat}(f)\leq\operatorname{relcat}(f)\leq\operatorname{secat}(f)+1$. D-EH conjecture. If $f$ admits a homotopy retraction, then

$$\operatorname{secat}(f)=\operatorname{relcat}(f).$$

Doeraene and El Haouari proved the displayed bounds, and the conjecture concerns exactly when the possible difference of one disappears. The hypothesis cannot be dropped: the paper gives examples, including the Hopf map, where the two invariants differ.