$T$-polynomial convexity and holomorphic convexity

We compare the $T$-polynomial convexity of Guedj with holomorphic convexity away from the support of $T$. In particular we show an Oka–Weil theorem for $T$-polynomial convexity, as well as present a situation when the notions of $T$-polynomial convexity and holomorphic convexity of $X\setminus \text{Supp }T$ coincide in the context of complex projective algebraic manifolds.