Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness

Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space $X$ and a dcpo $P$ equipped with its Scott topology, under what conditions on $P$ do the Isbell and Scott topologies on $C(X,P)$ agree?It was proved that, for a nonempty bicomplete continuous dcpo $P$, the Isbell and Scott topologies on $C(X,P)$ coincide for every core-compact space $X$ if and only if $P$ is bounded complete; for every compact core-compact space $X$ if and only if $P$ is conditionally bounded complete; and for every RW-space $X$ if and only if $P$ is a pointed continuous $L$-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.