# Large discrete subspaces in spaces of separately continuous functions

For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.

It is easy to see that for the real line $X=\mathbb R$ the space $SC_p(X\times X,X)$ contains a discrete subspace of size continuum.

Question: Is it true that for every infinite cardinal $\kappa$ there is a Tychonoff space $X$ of weight $\kappa$ such that the space $SC_p(X\times X,X)$ contains a discrete subspace of cardinality $2^\kappa$?

This question has an affirmative answer under GCH (in this case for $X$ we can take an ordered field of weight $\kappa$ and cardinality $2^\kappa$; such a field exists if $2^\kappa=\kappa^+$). So, the question actually asks what happens in ZFC.