[FOM] Cook continuum

[pax0 at seznam.cz]

Does anyone know what is the status of existence of Cook continuum
(non-empty compact connected space, not a singleton and for every 
subcontinuum every restricted continuous map
is either constant or identity)
to the axioms of ZFC/large cardinals?
Is it provable or independent?Where can I find proof of it?
Thank you, Jan Pax
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