 $X$ is not a separable submetrizable, i.e.($iw(X)>\omega$) $X$ has not a countable injective weight.
 There is a Baire isomorphism 1class between $X$ and a separable metrizable space $Y$.
Now take any uncountable $Q$set $Y\subset R$ and let $X$ be the onepoint compactification of a discrete space of cardinality $Y>\omega$.

$\begingroup$ Then $X$ and $Y$ is Borel isomorphic (1  class), but $X$ and $Y$ is not Baire isomorphic ! So the question remains open ! $\endgroup$ Nov 8 '17 at 6:33