The split field-redefinition conjecture for direct products of CYMH gauge theories

Let $N$ be a smooth manifold such that its tangent bundle admits a CYMH GT, and let $K\to S$ be a Lie algebroid bundle over a smooth manifold $S$ which also admits a CYMH GT. Set

$$E\coloneqq \mathrm{T}N\times K\to N\times S.$$

A field redefinition is understood to be specified by a valid parameter of the corresponding type. Split field-redefinition conjecture. If there is a field redefinition such that the direct product of CYMH GTs on $E$ is pre-classical or classical, then there is also a field redefinition with respect to a parameter of the form

$$\lambda=\lambda^N\times\lambda^K$$

such that the direct product is pre-classical or classical, respectively, where $\lambda^N\in\Omega^1(N;\mathrm{T}N)$ and $\lambda^K\in\Omega^1(S;K)$ are valid parameters for field redefinitions of the two factors. The conjecture is intended to reduce questions about direct products to separate field redefinitions of their tangent-bundle and Lie-algebroid-bundle factors; its status is open in the source, which presents it as a personal expectation rather than an established result.