Let $M$ be a (compact, let's say) Riemannian manifold, $V$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some potential $V \in \Gamma^\infty(M, \mathrm{End}(V))$.

Assume that $\nabla$ is metric with respect to the fiber metric on $V$ and that $V$ is symmetric. Then we have the inequality $$ |p_t(x, y)| \leq e^{t v} p_t^\Delta(x, y),$$ where $p_t$ is the heat kernel of $L$, $p_t^\Delta$ is the heat kernel of the Laplace-Beltrami operator on functions and $v \in \mathbb{R}$ is some constant such that $v < V$, as is proven by Hess, Schrader and Uhlenbroch in "Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds".

Is a similar inequality true for non-symmetric potentials $V$ and non-metric connections $\nabla$, and if yes, is this available in the literature?

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some potential $V \in \Gamma^\infty(M, \mathrm{End}(\mathcal{V}))$.

Assume that $\nabla$ is metric with respect to the fiber metric on $\mathcal{V}$ and that $V$ is symmetric. Then we have the inequality $$ |p_t(x, y)| \leq e^{t v} p_t^\Delta(x, y),$$ where $p_t$ is the heat kernel of $L$, $p_t^\Delta$ is the heat kernel of the Laplace-Beltrami operator on functions and $v \in \mathbb{R}$ is some constant such that $v < V$, as is proven by Hess, Schrader and Uhlenbroch in "Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds".

Is a similar inequality true for non-symmetric potentials $V$ and non-metric connections $\nabla$, and if yes, is this available in the literature?