# Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-valued map.

Problem S. Does each usco map $$\Phi:M\multimap K$$ from a compact metrizable space $$M$$ to a compact Hausdorff space $$K$$ admit a Borel selection?

Looking at the literature I found a lot of results addressing this problem. For example the classical Kuratowski-Ryll-Nardzewski Theorem implies the affirmative answer to Problem S if $$K$$ is metrizable. By Hansell, Jayne and Talagrand (1985), the answer to Problem S is affirmative if $$K$$ is fragmenable. On the other hand, by the result of Graf or Cascales-Kadets-Rodrigues, every usco map $$\Phi:P\multimap K$$ from a Polish space $$P$$ to a compact Hausdorff space $$K$$ has a universally measurable selection.

But to my surprise looking through the literature I cound not find an answer to Problem S (or I overlooked something?). All existing results imposing some restrictions on $$K$$ (like being fragmentable or linearly ordered) suggest that one should expect a counterexample. And indeed, there exists a counterexample under $$\neg CH$$:

Example. There exists an usco map $$\Phi:[0,1]^2\multimap K$$ to some Hausdorff compact space $$K$$ (namely, the square of split interval), which has a Borel selection if and only if the Continuum Hypothesis holds.

But what about a ZFC-counterexample to Problem S? What is known in this respect? Is the above Example known to specialists in the field?