Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ is algebraically closed and characteristic $0$, in fact I am content with $k = \mathbb{C}$). By a toric principal $G$-bundle $\mathcal{P}$ we mean $T$ acts on $\mathcal{P}$ lifting its action on $X_\sigma$ and the $T$-action and $G$-action on $\mathcal{P}$ commute.

My question is wether $\mathcal{P}$ is $T$-equivariantly trivial. By this, we mean the following: there exists a homomorphism $\phi: T \to G$ such that $\mathcal{P}$ is $T$-equivariantly isomorphic to $X_\sigma \times G$ where $T$ acts on this product diagonally by acting on $G$ via $\phi$ and left multiplication, that is, $t \cdot (x, g) = (t \cdot x, \phi(x)g)$.

It is well-known and not difficult to show that this is true when $G = GL(n)$ (that is, one has a toric vector bundle). As far as I know by a result of Biswas-Dey-Poddar (2015) which itself relies on an earlier result of Heinzner-Kutzschebauch (1995), this is true for when $X_\sigma$ is complex affine space. I was wondering if this is known for other affine toric varieties.