A Universe with large Continuum, global $\mathrm{\Sigma }$-Uniformization and a projective Well-Order of its Reals

We construct a model in which the continuum has size $\kappa$ for a regular cardinal $\kappa$ and in which the ${\mathrm{\Sigma }}_n^1$-uniformization property holds simultaneously for every $n\ge 2$. Additionally this model has a ${\mathrm{\Delta }}_3^1$- definable well-order of its reals.